in this video i'm going to focus on solving basic trigonometry problems now what i recommend that you do is pause the video and work out the problem first once you select your answer unpause it to see the solution and how to arrive there as well so let's start with this problem which of the following angle measures is equivalent to 60 degrees so basically what we need to do is convert degrees to radians so how can we do that the best way is to multiply by pi divided by 180 you want to set it up in such a way that the degree symbol cancels and so this becomes 60 pi over 180 now how do we simplify this for one thing we can cancel a zero sixty over one eighty is the same as six over eighteen and eighteen is six times three so now we could cancel a six and so sixty degrees is equivalent to pi over three and so that's a simple technique that you can use to convert degrees to radians so therefore b is the right answer number two which of the following represents an angle measure of negative 5 pi divided by 6 is it a b c or d well first let's go over the angles that you need to know this is 0 degrees 90 180 270. now you can deal with negative angles so if you're going in let's say the counterclockwise direction the angle measure is positive now if you travel in the clockwise direction the angle measure will be negative so in this case this is 0 degrees negative 90 negative 180 negative 270 if you're traveling in that direction so what we need to do is convert radians to degrees let's do that first so we have negative 5 pi divided by 6 so we need to multiply by 180 divided by pi so you want to do in such a way that pi cancels so we're left with negative 5 times 180 divided by 6. now we know that 18 is 3 times 6. so we could say that 180 is 6 times thirty and so we could cancel a six negative five times thirty is negative one fifty so that's the angle so this is zero negative ninety negative 180 so traveling in the clockwise direction negative 150 should be somewhere in this region therefore we can see that b is the right answer for this problem number three which of the following angles is coterminal to five pi over eight now before we could find the answer we need to know what a coterminal angle is so let's take 30 degrees for example 30 degrees is located in quadrant one so this is quadrant one quadrant two quadrant three quadrant four a coterminal angle is one in which it's going to land in the same position as the 30 degree angle that is in quadrant one and to find a coterminal angle you can either add or subtract 360 to it if you add 360 you're going to get 390. so 390 is coterminal to 30. so this is 90 180 270 360 390. so you land in the same position now if we did 30 minus 360 this will give us negative 330 which is also coterminal to 30. so starting from here this is going to be negative 90 negative 180 negative 270 negative 330. so these angles are coterminal because they have the same position in this graph so now let's focus on the problem which of the following angles is coterminal to 5 pi over 8 so there's two ways in which we can get the answer we can add 2 pi or we could subtract by 2 pi so let's start by adding two pi now keep in mind the reason why we're using two pi is because we have the angle and radians and it's important to understand that two pi is equal to 360 degrees pi is equal to 180 degrees so let's turn this into a fraction and we need to get common denominators so we have an 8 on the left side so we need to multiply the top and the bottom by 8. so this is going to be 5 pi over 8 and then 2 pi times 8 that's 16 pi so now that we have the same denominator we can add the numerators so 5 pi plus 16 pi is 21 pi so this is one answer which we do have so c is the correct answer now if we subtract it by two pi this is going to be five pi over 8 well first we need to multiply this value by 8 over 8. i'm getting ahead of myself so this is going to be 5 pi divided by 8 minus sixteen pi over eight and five minus sixteen is negative eleven so negative eleven pi over eight is also coterminal to five pi over eight so the other answers a b and d they're not coterminal to five pi over eight number four what is the length of the arc s in the figure below given an angle of 120 degrees and a radius of nine inches so the formula that we need to get this answer is this the arc length is equal to the angle and radians multiplied by the radius of the circle now we have the angle in degrees so we need to convert it to radians so we have 120 degrees and we need to multiply by pi divided by 180. so we could cancel the degree symbol and we could cancel a 0. so this is equal to 12 pi divided by 18 and 12 is 2 times 6 thirds i mean 18 is 3 times 6 and so we could cancel a six and so the angle in radians is two pi divided by three so the arc length is going to be two pi over three multiplied by a radius of nine inches now two times nine is eighteen and eighteen divided by three is six so the arc length is six pi now pi is 3.14159 so if you multiply 6 by that number this will give you 18.849 five four so we can round that and say it's 18.8 inches so e is the answer number five what is the value of sine x using the figures shown below so go ahead and try this problem now the first thing we need to do is we need to find the missing side and so we have a right triangle let's call this a b and c so we got to find c the side length of the hypotenuse and so we could use the pythagorean theorem c squared is equal to a squared plus b squared so a is 3 b is 4 and we need to calculate c 3 squared is 9 4 squared is 16 and 9 plus 16 is 25 so now we need to take the square root of 25 which is 5. so what we have is the 3 4 5 right triangle now there are some special triangles that you want to keep in mind the 345 triangle is simply one of them here are the other ones so in addition to this there's also the 5 12 13 triangle there's the 7 24 25 triangle and there's the 8 15 17 triangle now any ratio of these numbers will also work for example if i multiply the 3 4 5 triangle by 2 i get the 6 8 10 triangle or if i multiply by 3 i get the 9 12 15 triangle so those works as well now there are some other triangles that you may see they're rare but sometimes you might encounter them there's the 9 40 41 triangle and i've seen the 11 60 61 triangle in case you want to write those down now you need to be familiar with something called sohcahtoa so what does this expression mean well let's focus on the so part s stands for sine so sine of an angle it could be theta or it could be x is equal to opposite divided by the hypotenuse so that's the sine ratio o stands for opposite h is for hypotenuse so opposite to x is 4 and adjacent to x is 3 and 5 is the hypotenuse so let me just show you with a separate triangle so let's say this is the angle theta this is opposite to theta this is adjacent to it and this is the hypotenuse which is always across the 90 degree angle the hypotenuse is the longest side of the triangle now the next one k c a h c stands for cosine cosine theta is the ratio between the adjacent side of the triangle and the hypotenuse and the last one tangent which is associated with toa tangent is the ratio of the opposite side of the triangle to the adjacent side and so those are some things that you want to keep in mind so we're going to use that so in this example we want to find the value of sine x so sine x is going to equal the side opposite to it divided by the hypotenuse so opposite to x is four and the value of the hypotenuse is five so sine x is equal to four over five so therefore b is the right answer in this problem number six what is the value of secant x in the figure shown below so you need to be familiar with the reciprocal identities for instance cosecant is one over sine secant is one divided by cosine and cotangent is one over tangent and it's important to know that tangent is also equal to sine over cosine so those are some formulas that you may want to add to your list so in this case to find the value of secant first we need to find the value of cosine so using sohcahtoa we know that cosine is equal to the adjacent side divided by the hypotenuse so first we got to find the missing side now what type of triangle do we have is it a 3 4 5 triangle is it a 7 24 25 right triangle is it an 8 15 17 triangle is it a 9 40 41 or an 11 60 61 triangle this triangle is the 5 12 13 triangle and so that's the missing side and if you ever forget you can always use this formula a squared plus b squared is equal to c squared so a can be any one of these numbers so let's say a is five and we wish to calculate the value of b now let's say c is 13. 5 squared is 25 13 times 13 is 169. now 169 minus 25 is 144 and the square root of 144 is 12. so you can always find the missing side using this formula if you ever forget the special right triangles now let's focus on finishing this problem so adjacent to x is 12 and the hypotenuse is 13. so it's 12 over 13. keep in mind 5 is opposite to the angle x so now we have the value of cosine but our goal is to calculate the value of secant so secant which is 1 over cosine that's 1 over 12 divided by 13 which is 13 over 12. so the correct answer is answer choice d now if you wish to understand how this works here's one way in which you can simplify it so once you have this multiply the denominator the fraction by 13 and multiply the numerator by 13. so on the bottom these will cancel and so on the top of the fraction you're going to have 1 times 13 which is 13. and on the bottom you have a 12 left over so you get 13 over 12 which means that answer choice d is the correct answer to this problem number seven if sine x is equal to seven over 25 what is the value of tangent x if x lies in quadrant one so first let's draw a graph now keep in mind this is quadrant one quadrant two quadrant three quadrant four so what we need to do is draw a right triangle in quadrant one since that's where the angle is located in so this is going to be x now sine x is seven over twenty five and we know that sine is equal to the opposite side divided by the hypotenuse so opposite to x is seven because sine is equal to 7 over 25 and the hypotenuse is 25. now what type of right triangle do we have it's not a 5 12 13 or 3 4 5 triangle it's the 7 24 25 triangle so the missing side is 24. so now we can calculate the value of tangent x so based on sohcahtoa tangent is equal to the opposite side divided by the adjacent side so opposite to x is 7 and adjacent to it is 24 so tangent is 7 over 24 which means that e is the right answer number eight if tangent x is negative 8 over 15 what is the value of cosecant x if x is between three pi over two and two pi so this is zero this is ninety degrees which is pi over two this is pi that's 180 degrees and 3 pi over 2 is 270. 2 pi is coterminal to 0 degrees which is the same as 360 degrees so x lies somewhere in quadrant 4 between 3 pi over 2 and 2 pi so therefore we need to draw a triangle in quadrant four now x is a large angle and so x is measured from the positive x-axis now it's going to create a reference angle which we'll call theta now we can use the reference angle to calculate the value of tangent x now tangent is associated with the opposite side divided by the adjacent side and so tangent in this example is negative 8 over 15. so opposite to theta that's going to be negative eight and the adjacent side is 15. now this makes sense why this is negative because it's going in the negative y direction the 15 is along the positive x-axis so that's going to be positive 15. so here we're dealing with the 8 15 17 triangle so now we can calculate the value of cosecant x cosecant is 1 over sine and sine x is going to be the opposite side divided by the hypotenuse so sine theta which is going to be the same as sine x in this problem is equal to the opposite side which is negative 8 and the hypotenuse is 17. so if sine is negative 8 over 17 cosecant is the reciprocal of that so it's going to be negative 17 over 8. you just need to flip the fraction so the answer is this which correlates to answer choice a number nine if sine x is equal to five over seven what is the value of cosine x if x is between pi over 2 and pi so i'm going to use the pythagorean identities to get the answer for this problem you need to know that sine squared plus cosine squared is equal to one now there are some other identities that you need to be familiar with another pythagorean identity is this one one plus cotangent squared is equal to cosecant squared and also 1 plus tangent squared is equal to secant squared so make sure you know these three pythagorean identities now let's replace sine with five over seven and let's calculate the value of cosine five squared is twenty five seven times seven is forty nine and i'm going to replace one with 49 over 49 because 49 divided by 49 is one and i need to get common denominators so i'm going to move this and move it to the other side so 25 over 49 is positive on the left but it's going to be negative on the right side so cosine squared is 49 over 49 minus 25 over 49. 49 minus 25 is 24 and so cosine squared is 24 over 49. now let's take the square root of both sides the square root of cosine squared is cosine the square root of 49 is 7. now we need to simplify the square root of twenty four twenty four is four times six and the square root of four is two so the square root of twenty four is two square root six and the answer can be plus or minus now looking at our answer choices the only one that has 2 square root 6 over 7 is answer choice c so that has to be the answer now what if both answers were present so what if we had positive and negative two square root six over seven how can we determine which one is the correct answer well for one thing you need to know the signs of sine cosine and tangent for the four different quadrants so this is quadrant one quadrant two quadrant three and quadrant four now for a unit circle where the radius is one sine theta is equal to the y value cosine theta is equal to x and tangent theta which is sine divided by cosine that's y divided by x now y is positive above the x-axis it's positive in the first two quadrants y is negative below the x-axis that's it it's negative in quadrant three and four and sine is associated with the y value so therefore sine is positive in quadrants one and two and it's negative in quadrants three and four now cosine is associated with the x value and x is positive on the right side and it's negative on the left side so cosine is positive in quadrants one and four and it's negative in 2 and 3. now the last one tangent is y over x so therefore tangent is going to be positive whenever sine and cosine have the same sign that is they're both positive or both negative because a positive divided by a positive is a positive number a negative number divided by a negative number is a positive number now tangent is negative whenever sine and cosine have opposite signs if you divide a positive number by a negative number you're going to get a negative result or if you divide a negative number by a positive number it's still a negatively so so tangent is negative in quadrants two and four and it's positive and one and three so now x is between pi over two and pi so we're dealing with quadrant two and cosine is negative in quadrant two so therefore we know that the answer has to be c now for those of you who want to use right triangle trigonometry as opposed to pythagorean identities to solve this problem here's what you can do so we realize that x is in quadrant two it's between 90 and 180 so we need to draw the right triangle in quadrant two so that's the value of x so it's going to form a reference angle theta now we know that sine is equal to opposite divided by the hypotenuse and so sine is five over seven opposite to theta is five and hypotenuse is seven so let me draw the right angle now we need to find the value of the missing side so we have to use the pythagorean theorem to do so so a squared plus a b squared is equal to c squared we're going to say a is 5 and we're looking for the missing side b c is 7. 5 squared is 25 and 7 squared is 49 49 minus 25 is 24 so now we need to take the square root of both sides 24 is 4 times 6 and the square root of four is two so this is going to be two square root six now y is positive above the x axis and x is negative on the left side so this is going to be negative two square root six so now we can calculate the value of cosine cosine is equal to the adjacent side which is negative two square root six divided by the hypotenuse which is seven so as you can see this will give us the same answer which is answer choice c number 10 what is the exact value of cotangent x given the point negative 40 comma negative 9 which lies on the terminal side of the angle x so how can we find the answer and what exactly is meant by the terminal side of an angle now an angle is formed between two rays so here's the first ray and here's the second ring the first ray is known as the initial side the second ray is called the terminal side and it makes sense why it's called this way because the angle is measured from the initial side and it ends at the terminal side and it meets at a common endpoint known as the vertex so now you know what the terminal side of an angle is but how do we use this information to get the exact value of cotangent x well first we need to plot this point so first we need to travel 40 units to the left and then we need to go down by 9 units so this is negative 40 this is negative 9 and the hypotenuse of this triangle is the terminal side of the angle and along the x-axis we have the initial side so the angle x is measured from the initial side to the terminal side so that's x which i'm going to draw like this now this is going to form a reference angle theta and cotangent x and cotangent theta will give you the same result so keep that in mind now we still have a right triangle and this is a special right triangle it's the 9 40 41 triangle so therefore the hypotenuse is 41. the hypotenuse is always positive for these types of problems so now that we've completed the triangle all we need to do is calculate the value of cotangent but let's find the value of tangent first tangent x which is the same as tangent theta in this problem is equal to the opposite side divided by the adjacent side so opposite to theta is negative nine adjacent to it is negative forty negative nine divided by negative forty is positive nine over forty now cotangent is simply the reciprocal of tangent so if tangent is nine over forty cotangent is forty over nine so tangent and cotangent are positive in quadrant three so d is the right answer 40 divided by nine now one thing i do want to mention now we know that this is the terminal side so i'm going to write t for terminal side and when it says that the point lies on the terminal side the point is actually right here this is the point where x is negative 40 and y is negative nine so thus you can see why the point lies on the terminal side of angle x so i just want you to understand the problem visually so you know what to do in order to get the answer number eleven identify the quadrant in which sine z is less than zero and cosine z is greater than zero so the fact that sine z is less than zero tells us that sine is negative and cosine is positive now sine is associated with the y value and cosine is associated with the x value so in which quadrant is x positive and y is negative so x is positive towards the right y is negative as you go down so therefore we're dealing with quadrant four that's the answer so that's when x is positive and y is negative now perhaps you heard of the expression all students take calculus do you know what that means what do you think it means so the first word all tells us that everything is positive in quadrant one and it goes in order so all students s for students so in quadrant two sign is positive take that means tangent is positive in quadrant three and then calculus cosine is positive in quadrant four everything else is negative and so you can also use that expression to help you remember what's positive what's negative if you prefer to do it that way sine is negative in quadrants 3 and four cosine is positive in quadrants one and four so the answer occurs in quadrant four that's when sine is negative and that's when cosine is positive so therefore d is the right answer number 12 which of the following expressions is equivalent to sine negative pi over three now in order to find the answer you need to be familiar with the even and odd trigonometric functions by the way if you need to find more uh videos on trich check out my playlist it's my new trigonometry playlist you can search it on youtube or you could find it in this video so if you click that button on the right you should be able to access it if not you can check in the description section of this video to find it or at one of the end screens at the end of this video it should show up at the last 20 seconds of this video so let's focus on this problem now you need to know that sine of negative x is equal to negative sine x so sine is an odd function cosine negative x is equal to positive cosine x so cosine is an even function cosecant which is the reciprocal of sine that's an odd function so it's negative cosecant x secant which is the reciprocal of cosine is an even function so secant negative x is the same as secant x now the next one is tangent tangent is an odd function like sine so tan negative x is negative tangent x and cotangent is also an odd function cotan negative x is negative cotan x so make sure you know these properties now let's go ahead and finish this so we said that sine negative x is equal to negative sine x so x is going to be pi over 3. so sine negative pi over 3 has to equal negative sine pi over 3 based on that identity so therefore d is the correct answer number 13 which of the following expressions is equivalent to sine pi over five now how can we find the answer if we don't have access to a calculator you need to be familiar with the cofunction identities so for instance sine theta is equal to the cofunction of sine is cosine so it's cosine 90 minus theta and cosine theta is sine 90 minus theta now tangent theta the cofunction of tangent is cotangent so this is going to be cotangent 90 minus theta and the same is true for cotan cotan theta is going to be tangent 90 minus theta now secant the cofunction of secant is cosecant so this is going to be cosecant 90 minus theta and for cosecant theta it's secant 90 minus theta so let's go over some examples of this so for instance sine of 30 degrees is equal to cosine 90 minus 30 which is basically cosine of 60 degrees tangent of 40 degrees is equal to the cofunction cotangent 90 minus 40 degrees and so 90 minus 40 is 50 so this is equal to cotangent of 50 degrees secant of 20 degrees is equal to cosecant of 90 minus 20 degrees so 90 minus 20 is 70. so that's a quick and simple way in which you can identify the equivalent cofunction expression so now we have our angle in radians not in degrees and 90 is equivalent to pi over 2. keep in mind that pi is equal to 180 half of pi is pi over 2 and half of 180 is 90. so 90 is the same as pi over 2. so sine pi over 5 has to be equal to cosine 90 minus pi over 5 or pi over 2 minus pi over 5. so we need to subtract these two angles so we need to get common denominators so i'm going to multiply pi over 2 by 5 over 5 and pi over 5 i'm going to multiply that by 2 over 2. so 5 times 2 is ten so this is going to be five pi over ten minus two pi over ten and five minus two is three so this is equal to three pi over ten so this is the answer it's cosine three pi over ten which means d is the right answer now if you have access to a calculator you can simply plug everything in but make sure your calculator is in radian mode not in degree mode so if we type in sine pi over 5 this is equal to 0.588 so that's the answer that we need to get so let's start with answer choice a now let's type in sine 7 pi divided by 10. so this is equal to positive 0.809 now let's move on to b cosine 6 pi divided by five so this is equal to negative point eight zero nine and then part c sine negative four pi divided by five this is negative point five eight eight it's close but it's not the answer now for d cosine 3 pi over 10 this is positive 0.588 so this is the answer we're looking for therefore d is the right answer choice number 14 which of the following expressions is equal to cosine negative 150 so now let's talk about how we can do this without using a calculator or unit circle or special reference triangle so what do you think we can do well for one thing we can start with the cofunction identity because we do have some values in sine so we know that cosine theta is equal to sine 90 minus theta so we have cosine of negative 150 so therefore this is equal to sine 90 minus theta which is negative 150 so 90 minus negative 150 that's the same as 90 plus 150 so that's sine 240. now e would be correct if there wasn't a negative sign so therefore we could eliminate answer choice e now what else can we eliminate now let's check out the even odd properties of cosine so cosine is uh an even function we know that cosine negative x is just cosine x so what this means is that cosine negative 150 is equal to cosine positive 150 now c is close to that but there's a negative sign in front of this so we could eliminate answer choice c now what about the rest a b and d what can we do the next thing we could do is look at the sine of cosine is it positive or is it negative depending on certain quadrants so let's make two graphs so if we have a positive angle this is 0 90 180 and 270. and if we're dealing with negative angles this is zero negative 90 negative 180 negative 270. so negative 150 is located in quadrant three so that's for this angle and cosine is negative in quadrant three because x is negative in quadrant three we're dealing with the left side so we need a negative value for cosine now looking at answer choice b 30 is over here that's in quadrant one so cosine 30 is equal to a positive value which means we can eliminate answer choice b and then if we look at sine 60 60 is also in quadrant one and both sine and cosine are positive in quadrant one so we can eliminate answer choice d now 210 is in quadrant three and so cosine is negative in quadrant three so a has to be the answer another way in which you can confirm that a is the answer is using the periodic properties of trigonometric functions so for instance sine of 60 is equal to the sine of a coterminal angle such as 60 plus 360 which is 420 or if you do 60 minus 360 which is negative 300. so these are periodic properties of the trig functions so cosine of an angle is equal to cosine of that same angle plus or minus 2 pi or 360 degrees so we have cosine of negative 150 that's going to be equal to cosine negative 150 plus 360. and so negative 150 plus 360 that's 210 and so that shows that a is indeed the correct answer now for those of you who might be taking a test or a final exam or midterm and if your teacher allows you to use the calculator i would highly recommend that you use it for a problem like this because it'll make the whole problem a lot easier so first make sure your calculator is in degree mode if you don't see a pi next to a number then the angle is in degrees so let's plug in cosine negative 150. this is equal to negative square root 3 over 2. now if we plug in cosine 210 we're going to get the same answer negative square root 3 over 2. now my calculator just gives it to me in that form your calculator might write it like this it might display it as negative 0.866 and so for answer choice a is the same cosine 30 is positive square root 3 of 2 but you might see it as positive 0.866 a negative cosine 150 this is equal to positive 0.866 sine 60 is also positive 0.866 a negative sign 240 that 2 is positive 0.866 so those are all equal to each other the only one that's different is a but a is the answer 15 which of the following answer choices is equivalent to the expression shown below so 1 plus sine squared of 40 degrees plus sine squared of 50 degrees is that equal to cosine 45 sine 90 2 or sine 10 so how can we find the answer to this problem and what if we don't have access to a calculator because if we have access to a calculator you could just type in the whole thing get the decimal value and then type in each answer choice and see which one matches the answer uh above so that technique can always work for these types of problems if you have access to a calculator but now let's assume that we don't have access to a calculator what should we do well for one thing we can use the cofunction properties of sine so recall that sine theta is equal to cosine 90 minus theta so sine 50 is equal to cosine 90 minus 50 and 90 minus 50 is 40. so therefore we could say that sine 50 is equal to cosine 40. so therefore if we square both sides we could say that sine squared of 50 is equal to cosine squared of 40. so therefore let's replace sine squared 50 with cosine squared 40. now we can use another identity and that is the pythagorean identity for sine and cosine sine squared plus cosine squared is equal to one if the angle is the same which in this case it is so we can replace this part sine squared plus cosine squared of one so the answer becomes one plus one which is equal to two so therefore c is the right answer in this problem number sixteen what is the exact value of cosine 60 don't use the calculator that's just going to be too easy so there's two other ways to find the answer if you don't have access to a calculator you can use the 30 60 90 reference triangle or you could use the unit circle but let's focus on a reference triangle first so let's say this is 30 and this is 60. across the 30 the side value is across the 60 the side length is equal to the square root of three across the 90 the hypotenuse is two so if you memorize this triangle it can help you to find the value of like sine 30 or cosine 60 and things like that so recall that cosine theta is equal to the adjacent side divided by the hypotenuse based on sohcahtoa so we want to evaluate cosine of 60 so we need to focus on the 60 degree angle now 2 is clearly the hypotenuse that's always going to be the case opposite to 60 is the square root of 3 so 1 must be the adjacent side so it's adjacent over hypotenuse that means it's 1 divided by 2. and so that's how you can find the exact value of cosine 60 which means that answer choice b is the correct answer now let's say if you have access to the unit circle or if you want to commit the unit circle to memory by the way if you need a copy of the unit circle go to google images and type in unit circle and you should see a bunch of it pop up now in the unit circle at an angle of 60 degrees or pi over 3 you'll see these values one-half comma square root three over two now for a unit circle keep in mind that sine theta is equal to y and cosine theta is equal to x well in this example theta is sixty so cosine 60 is equal to the x value which is one half and sine 60 for example is equal to the y value which is the square root of 3 over 2. so that's how you can use the unit circle to evaluate certain trigonometric functions with their angles so in this case b is the right answer number 17 what is the exact value of sine pi over 4 first let's convert the angle from radians to degrees so it's pi over 4 and we're going to multiply by 180 divided by pi so basically we're replacing pi with 180 and so we're going to get 180 over 4 because pi equals 180 so what's 180 divided by so how can we find the answer for that well if you don't have access to a calculator you could use long division four goes into 18 four times four times four is sixteen and eighteen minus two is i mean eighty minus sixteen is two and then bring down the zero four goes into 25 times and so the remainder is zero so 180 divided by four is 45. so we need to find the value of sine 45 degrees so we need to use the 45-45-90 reference triangle so this is another one that you want to commit to memory across the 45-degree angles it's one the hypotenuse is the square root of two because the angles are the same the side lengths have to be the same so now recall that sine is equal to opposite divided by hypotenuse so let's pick one of the angles opposite to 45 is one and the hypotenuse is clearly square root two now we need to rationalize the denominator so we're going to multiply the top and the bottom by the square root of two so one times the square root of 2 is the square root of 2. the square root of 2 times the square root of 2 we know that 2 times 2 is 4 so that becomes the square root of 4 which is equal to 2. so sine 45 is equal to the square root of 2 divided by 2 which means c is the correct answer 18 what is the exact value of tangent 30 so go ahead and try that problem so let's use the 30-60-90 triangle so across the 30 is 1 across the 60 is the square root of 3 across the 90 is 2. so recall that tangent of theta is equal to the opposite side divided by the adjacent side and so we're focusing on the 30 degree angle opposite to 30 is one and adjacent to 30 is the square root of 3. so we need to rationalize it so this becomes the square root of 3 and then the square root of 3 times itself is the square root of 9 which is 3. so tangent 30 is root 3 divided by 3. so b is the right answer now if you wish to use a unit circle to get the answer here's what you can do so first you need to locate the 30 degree angle on the unit circle so at 30 degrees you'll see that x is the square root of three over two and y is one half now recall that tangent is sine divided by cosine it's y divided by x so sine 30 is one half cosine 30 is the square root of three over two now if we multiply the top and the bottom by two we can get rid of some fractions so on top this is going to be equal to one on the bottom the twos will cancel and we're going to get the square root of three and then we can rationalize it giving us the same answer square root 3 divided by 3. so b is the right answer keep in mind that cosine is associated with the x value and sine is associated with the y value now as a side question what is the value of tangent 90 if you use the unit circle you need to know that at 90 degrees x is zero y is one and so tangent ninety is going to be y divided by x the y value is one x is zero so what's one divided by zero anytime you get a zero in the denominator that means that it's undefined so if you were to type in tangent of 90 in your calculator make sure it's in degree mode it's going to give you an error and so when you see that it means that the answer is undefined that means that you have a zero in the denominator of a fraction so just watch out for that number 19 what is the reference angle of 290 degrees well first need to realize that the reference angle is an acute angle between 0 and 90. so the first thing you need to do is you need to determine what quadrant the angle is in so let's say if we have an angle in quadrant one let's say the angle is 60. the reference angle is the angle between the x-axis and the terminal side of the original angle so in quadrant one the reference angle is the equal to or the same as the angle in quadrant one so in this example it is 60 degrees now what if we have an angle that's in quadrant two let's say it's 150 degrees what's the reference angle so the reference angle is between the x-axis and the terminal side of the original angle so it's going to be the angle inside here so going from the positive x-axis to the negative x-axis that's 180 less 150 so the angle on the inside must be 30. therefore to calculate the reference angle for an angle in quadrant 2 it's going to be 180 minus the angle in quadrant two so in this example it was 180 minus 150 which is 30. now what if the angle is in quadrant three what's the formula so let's say if the angle is 240 degrees how can we calculate the reference angle so the reference angle is between the x-axis and the terminal side so that's this angle here and we know this is 180 this is 240 so the difference between those two values is the reference angle which is 60. so for an angle in quadrant three the reference angle is going to be the angle in quadrant three minus 180 so was 240 minus 180 and that gave us 60. now let's consider an example in quadrant four so let's use 330 as an example so we need to find the angle between the x-axis and the terminal side so that's this angle here now a full rotation around the circle is 360. so less 330 therefore this angle has to be 30 degrees so to calculate the reference angle for an angle in quadrant 4 it's always going to be 360 minus the angle in quadrant 4. so those are some formulas that you want to keep in mind if you wish to calculate the reference angle now in our example the angle is 290. so the first thing we need to do is identify what quadrant 290 is located in so this is 0 90 degrees 180 270 and 0 is the same as 360. those two angles are coterminal to each other so 290 is located in quadrant four so therefore we need to use this equation so the reference angle is simply going to be 360 minus the angle in quadrant 4 which is 290. 360 minus 290 is 70. so that's the reference angle between the x-axis and the terminal side it's 70 degrees which means e is the answer so now you know how to find a reference angle of any angle number 20 what is the exact value of cosine 210 and do not use a calculator so we know this is 0 90 180 this is 270. so 210 is located in quadrant three so this is an angle of 210. now what we need to do is calculate the reference angle so for an angle in quadrant three the reference angle is the angle in quadrant three minus 180 so that's going to be 210 minus 180 and so the reference angle is 30 degrees so let's turn this into a triangle now recall the 30 60 90 triangle this is 30 this is 60 across the 30 is one across the sixties root three across the ninety is two so now in quadrant three across the 30 is also one but since we're going along the negative y axis this is going to be negative one now this is the 90 degree angle so the hypotenuse has to be two so over here is the 60 degree angle so across the 60 is the square root of three now we're going in the negative x direction so that has to be negative square root 3. so now we can evaluate cosine of 210 using a 30 60 90 reference angle and that's why it's important to calculate the reference angle because it can help you to find the exact value of these functions if you don't have access to a calculator or if you don't have access to a unit circle so cosine of 210 which is equal to cosine 30 but cosine 30 is positive cosine 210 is negative so it's really negative cosine 30. and so that's going to equal the adjacent side divided by the hypotenuse so adjacent to the 210 angle is negative square root 3 and hypotenuse is 2. so cosine 210 is negative square root three over two cosine thirty is positive square root three over two everything here is positive so that's why cosine 210 is negative cosine thirty and also the fact that 210 is in quadrant three and you're dealing with cosine you know cosine is negative in quadrant three so it can't be d e or a so it has to be between b and c but for this problem we know the answer is c you