# Thread: I am given a side length and an angle, find the remaining side lengths.

1. ## I am given a side length and an angle, find the remaining side lengths.

Hi all,

The problem is: A right triangle has a hypotenuse of length 15 meters and makes an angle of 40 degrees with side S1. What are the lengths of the sides of the triangle, S1 and S2.

To solve, I set up a right triangle and marked the hypotenuse with a length 15, but I don't know where to take it from here.

I tried using the sine law: Sin (A)/a = Sin(B)/b but do not think I am applying the formula correctly.

2. ## Re: I am given a side length and an angle, find the remaining side lengths.

Originally Posted by Odonsky
Hi all,

The problem is: A right triangle has a hypotenuse of length 15 meters and makes an angle of 40 degrees with side S1. What are the lengths of the sides of the triangle, S1 and S2.

To solve, I set up a right triangle and marked the hypotenuse with a length 15, but I don't know where to take it from here.

I tried using the sine law: Sin (A)/a = Sin(B)/b but do not think I am applying the formula correctly.
This is a right triangle. There is no need to use the law of sines.

You are given the hypotenuse length and the angle between S1 and the hypotenuse.

From this and the definition of the cosine function

cos(angle between S1 and hypotenuse) = S1/hypotenuse so

cos(40deg) = S1/15

you can solve this for S1. You can then use either sin(40deg) or the Pythagorean theorem to solve for S2.

3. ## Re: I am given a side length and an angle, find the remaining side lengths.

Thank you romsek. Initially, I was unsure which side to label as S1 and S2, now it appears that it does not matter since we are given the hypotenuse.

In solving this problem, I get: cos(40) = S1/15 which equals 11.5

Now to solve for S2 I used Pythagorean theorem as you pointed out and get 9.63

4. ## Re: I am given a side length and an angle, find the remaining side lengths.

With a Right Triangle, you are always given a "gimme" angle, the 90* is a freebie, so with the one angle that they spell out for you of "40*", you can find the third with simple math before deciding which trig to use to get all the angles. 180-90-40=theta. You have the options of AAA then AAS or Law of Sines that you were trying to use but they all require previous know how on how to apply the formulas.

AAS: a^2sin(B)sin(c)/2sin(A), {in Radians}, a^2sin(B)sin(pi-A-B)/2sin(A), all letters would be rearranged for the known side, in this case "a"
Law of Sines: for b=(a)*sin(B)/sin(A), all letters would be rearranged for the known side, in this case "a"

In your case, the hypotenuse is the radius of the circle, the S1 leg is along the X Axis and the angle given is the adjacent angle, adjacent to the X Axis and the hypotenuse, as the X is known and the r is known, use the Cah (Cosine, adjacent, hypotenuse) from Soh Cah Toa to decide which formula to use, the Cosine formula of X/r is ready for computation to find the vertical leg length, once known, use the Sine formula, Y/r to verify and the Tangent, Y/X to double test the hypotenuse and all related values can be double checked for accuracy if needed.

From a beginner, test all permutations before deciding on final answer, if time allows. Please make corrections if needed.