in this video we're going to talk about how to solve right triangles so we're given the length of a side and an angle we want to calculate every other angle and every other side of the right triangle now the three angles of a right triangle must add up to 90 i mean 180 degrees this is 30 this is a right angle which is 90. so the missing angle x is what we need to find so 180 is going to equal the sum of those values x plus 90 plus 30. 90 and 30 is 120 and subtracting both sides by 120 will give us 60. so the missing angle is 60 degrees now what i'm going to do is label everything so let's replace x with 60 degrees we're going to call this angle a angle b is 60 and angle c that's going to be the right angle which is 90. now across angle c we have side c which is the hypotenuse of the right triangle across angle b is side b which is 12 and across angle a that's going to be lower case a which we need to calculate so how can we calculate a and c in this example well we need to use trigonometry perhaps you're familiar with this expression sohcahtoa the so part tells us that sine sine of the angle let's say sine of 30 is equal to the side that's opposite to it the o stands for opposite opposite to 30 is a divided by the hypotenuse the hypotenuse is always across the right angle the hypotenuse is c in this case so sine 30 is equal to a over c the next one k tells us that cosine of the angle which is 30 is equal to the adjacent side adjacent to 30 is b a is opposite to 30. and c is the hypotenuse so cosine 30 is going to equal the adjacent side b divided by the hypotenuse which is c now we could use this formula because we know the value of b b is 12 and so we need to calculate c cosine 30 is the same as cosine 30 over one what i'm going to do is cross multiply 1 times 12 is 12 and that's going to equal c times cosine of 30 degrees now you could use your calculator to figure this out cosine 30 degrees is equal to the square root of 3 over 2 which is 0.866 you can also look it up online or use a reference triangle to evaluate cosine 30. i do have other videos on that that really explains how to do that so i'm not going to discuss it much in this video now what i'm going to do is multiply both sides by 2 over the square root of 3. on the right side the 2's will cancel and the square root of 3's will cancel so on the left we have 2 times 12 which is 24 divided by the square root of 3 and that's going to equal c i'm going to rationalize this the square root of 3 times the square root of 3 is the square root of 9 which is equal to 3 and 24 divided by 3 is 8. so c is equal to 8 square root 3. now let's just clear away a few things because there's just a lot of stuff on the board right now so now that we have the value of c and b we could use the pythagorean theorem to calculate the value of a so we could use this formula a squared plus b squared is equal to c squared b is 12 and c is 8 square root 3. 12 squared or 12 times 12 that's going to be 144 8 squared 8 times 8 is 64. the square root of 3 squared is just 3. so 64 times 3 6 times 3 is 18 so 60 times 3 is 180 4 times 3 is 12 180 plus 12 gives us 192. now 192 minus 144 is 48 so a squared is 48. taking the square roots of both sides we get a is the square root of 48 and 48 is 16 times 3 the square root of 16 is 4. so a is 4 square root 3. so now we have everything we have all three sides of the right triangle and we know all three angles so that's what you need to do in order to solve a right triangle you need to solve for everything any angle or side length that's missing you gotta calculate its value so that's it for this example problem now let's try another one so let's say we have another right triangle we're going to say this is angle a angle b angle c and let's say that side a is equal to 21 and c side c the hypotenuse is 29. we don't know what b or angle a or angle b is equal to go ahead and solve for this right triangle now the first thing that i would do is i would calculate the missing side length because i have the two of the silence so we could use the pythagorean theorem for this a squared plus b squared is equal to c squared so a is 21. b is what we're looking for c is 29. 21 squared is equal to 441. 29 squared that's equal to 841 so subtracting both sides by 441 we're gonna get 400 and now we need to take the square root of 400. 400 is 4 times 100 the square root of 4 is 2 the square root of 100 is 10 2 times 10 is 20. so the missing side length is 20 units long so that's the answer for that and now let's just clear away this stuff and let's move on to the next thing that we need to do which is we need to calculate one of the angles angle a or angle b let's focus on angle a now let's go back to the sohcahtoa expression we talked about sine and cosine but we didn't talk about tangent so let's use tangent tangent of angle a is going to be equal to so t is for tangent o is for opposite a is for adjacent so it's equal to the opposite side divided by the adjacent side opposite to angle a is side a and adjacent to angle a is b the hypotenuse is always across the right angle so c is the hypotenuse so relative to a this is the opposite side this is the adjacent side and this is the hypotenuse but just in case you're curious relative to b this is the opposite side relative to b so therefore this is the adjacent side relative to b and this will always be the hypotenuse so just keep that in mind it changes if your perspective changes from a to b but relative to a angle a b is the adjacent side side a is the opposite side so we have tangent of angle a is equal to side a which is 21 divided by side b which is 20. now in order to get the angle we need to use the inverse tangent function or the arc tangent and whenever you use the inverse function you need to switch x and y in this case you need to switch the angle and the value so if tangent of angle a is equal to a value of 21 over 20 then arc tangent of the value is equal to the angle and that's how we can calculate the missing angle of a right triangle in this particular example go ahead and use your calculator and type in arc tangent 21 over 20 and you should get 46.4 degrees as angle a by the way make sure your calculator is in radian mode i mean i take that back make sure your calculator is in degree mode and not in radian mode if you're getting something different now in order to calculate angle b it's going to equal 180 minus the other two angles angle a and angle c so it's 180 minus 46.4 minus 90. because the three angles of a right triangle must add up to 180 so it's going to be 43.6 that is the value of angle b now because you're dealing with right angles to get angle b you could just do 90 minus 46.4 which will be faster but just in case you're solving a triangle that is not a right triangle you're going to have to use this one but since this video is about right triangles you can just do 90 minus 46.4 but that's it for this example so now you know how to solve right triangles don't forget to subscribe to this channel and hit that notification bell while you're at it thanks again for watching