in this video we're going to talk about how to find the exact value of trig functions so the first thing we need to talk about are the 30-60-90 right triangle you need to be familiar with this special triangle across the 30 degree angle is 1 across the 60 degree angle is the square root of 3 and the hypotenuse is 2. now the second triangle we need to be familiar with is the 45 45 90 triangle so across the 45 degree angles the sides are equal to one across the 90 degree angle it's the square root of two and you also need to be familiar with something called sohcahtoa this tells us the equations for the sine ratio the cosine ratio and the tangent ratio so let's say we want to find the exact value of sine 30 degrees so we would use this formula i mean that part of sohcahtoa it tells us that sine 30 is equal to the opposite side divided by the hypotenuse opposite to 30 is 1 and the hypotenuse of that triangle is 2. so the exact value of sine of 30 degrees is one over two now let's work on some other examples let's say we want to find the value of cosine five pi over six so what is the exact value of that particular trig function well it helps to convert the angle from radians to degrees and to do that multiply it by 180 over pi pi is equal to 180 degrees note that the unit pi will cancel so first let's divide 180 by 6. 18 divided by 6 is 3. so 180 divided by 6 that's going to be 30. and 30 times 5 is 150 so cosine 5 pi over 6 is the same as cosine 150 degrees now you might be wondering how do we evaluate cosine 150 because we don't have that in our two special triangles so what we need to do is plot 150 degrees and then create one of these two triangles from it so starting from the positive x-axis this is 90 and this is 150. so now let's create our reference triangle notice that 150 is 30 degrees away from the negative x-axis which is 180 degrees away from the positive x-axis so it creates a reference angle of 30. and if we draw a right triangle we know that this is going to be 90 and this is 60. so now we can put these values in that triangle we just need to incorporate the negative signs depending on what quadrant we're dealing with so this is quadrant one quadrant two quadrant three and quadrant four in quadrant two x is negative so this is going to be negative square root three and in that quadrant y is positive so this is just going to be 1 and hypotenuse is always going to be positive so now we have everything that we need to know in order to get the answer so cosine 150 degrees is equivalent to cosine of the reference angle or cosine of 30 in quadrant two and using sohcahtoa cosine is going to be equal to the adjacent side divided by the hypotenuse so that's what's symbolized by a and h so adjacent to 30 is negative square root 3 and the hypotenuse is 2. so cosine of 5 pi by 6 which is the same as cosine 150 that's equal to negative square root 3 over 2 and that's the exact value now here's another example let's try tangent pi over six so feel free to pause the video if you want to try that problem now let's convert it from radians to degrees now remember pi is equal to 180 so pi over 6 is 180 divided by 6 which is 30 degrees so if you simply replace pi with 180 you can quickly change it to degrees so what is tangent 30 well based on sohcahtoa tangent is equal to the opposite side divided by the adjacent side so opposite to 30 is one adjacent to 30 is the square root of three now we don't want to leave our answer like this we want to rationalize the denominator and we can do that by multiplying the top and the bottom by the square root of 3. the square root of 3 times the square root of 3 that's going to be the square root of 9 and the square root of 9 is three so tangent pi over six is equal to positive square root three over three now let's move on to our next example so next up we're gonna try cosine of 240 degrees take a minute and work on that problem so let's begin by drawing a reference triangle so this is 90 180 270 so we need to go back a little so 90 180 so 240 should be somewhere right there why does that keep happening this computer has issues so let's draw the reference angle so this is 90 180 then the difference between 240 and 180 gives us our reference angle which is 60. and so 30 is right here now we know that across the 30 degree angle is one and in quadrant three both x and y are negative so this is going to be negative 1 across 60 is root 3 but it's going to be negative square root 3 and the hypotenuse is going to be positive 2. so cosine 240 is going to be the same as cosine 60 but in quadrant 3. so using adjacent over hypotenuse adjacent to our reference angle 60 that's going to be negative one and the hypotenuse is two so cosine of 240 is equal to negative one-half now let's consider tangent of negative 45 degrees so once again let's draw a reference triangle now a positive angle will in order to draw a positive angle you need to move counterclockwise from the positive x-axis in order to draw a negative angle we need to move clockwise from the positive x-axis so we're going to travel 45 degrees this way which means our reference angle is 45 degrees but in quadrant 4. now we know that across the 45 degree angle it's going to be one but in quadrant four x is positive but y is negative and the hypotenuse is going to be positive square root 2. so now using sohcahtoa it's going to be the opposite side we'll use this 45 degree angle opposite to that is negative 1 divided by the adjacent side which is 1. so tangent of negative 45 is simply equal to -1 now let's work on one more problem and that's going to be sine of ten pi over three feel free to pause the video if you want to try that let's begin by converting the angle from radians to degrees and let's replace pi with 180 so let's divide 180 by 3 180 divided by 3 well we know 18 divided by 3 is 6 so 180 divided by 3 is 60 and then if we multiply 10 by 60 this will give us an angle of 600 degrees so sine of 10 pi over 3 is equal to sine of 600 degrees so this is 90 180 270 360 450 540 and if we go 60 degrees more we'll end up at 600 so that's an angle of 600 degrees so note that 600 is the same as 90 180 240 because this is 60 degrees more so 600 and 240 they land at the same location they are coterminal angles keep in mind a full circle is 360. so if you were to simply subtract 600 degrees by 360 you get the coterminal angle of 240 and that's what you need to do when you're dealing with very large angle or even negative angles so if you get an angle that's greater than 360 like 500 or 600 subtract it by 360. let's see if you have a super large angle like a thousand keep subtracting by 360 until the angle is between 0 and 360. now let's say if you have a negative angle like cosine negative 800 instead of subtracting it by 360 you want to add by 360 until you get an angle between 0 and 360. so you need to add 360 multiple times so i'm going to take a calculator just to demonstrate that so if we take negative 800 and add 360 to it it's going to be negative 440. if we add 360 again it's going to be negative 80. and then if you add it one more time that's 280. 280 is the angle that i would use so these are coterminal angles and they will yield the same value so sine of 600 is the same as sine of 240. so now let's create our reference angle at 240. so this is 90 180 60 more is 240 so our reference angle is 60 degrees if this is 60 we know this is 30 and let's draw our 90 degree angle so we're dealing with quadrant three which means x and y are both negative so across 60 that's going to be the square root of 3 and across 30 that is going to be 1 but negative 1. and then let's put our hypotenuse so using sogotoa we know that sine is equal to opposite over hypotenuse so our reference angle is 60 opposite to 60 is negative square root 3 and the hypotenuse is 2. so sine 10 pi over 3 is equal to negative square root 3 over 2. so now you know how to find the exact value of trigonometric functions