CCSS Math: HSG.SRT.C.8
Review right triangle trigonometry and how to use it to solve problems.

What are the basic trigonometric ratios?

sin(A)=\large\sin(\angle A)=oppositehypotenuse\large\dfrac{\blueD{\text{opposite}}}{\goldD{\text{hypotenuse}}}
cos(A)=\large\cos(\angle A)=adjacenthypotenuse\large\dfrac{\purpleC{\text{adjacent}}}{\goldD{\text{hypotenuse}}}
tan(A)=\large\tan(\angle A)=oppositeadjacent\large\dfrac{\blueD{\text{opposite}}}{\purpleC{\text{adjacent}}}
Want to learn more about sine, cosine, and tangent? Check out this video.

Practice set 1: Solving for a side

Trigonometry can be used to find a missing side length in a right triangle. Let's find, for example, the measure of ACAC in this triangle:
We are given the measure of angle B\angle B and the length of the hypotenuse\goldD{\text{hypotenuse}}, and we are asked to find the side opposite\blueD{\text{opposite}} to B\angle B. The trigonometric ratio that contains both of those sides is the sine:
sin(B)=ACABsin(40)=AC7B=40,AB=77sin(40)=AC\begin{aligned} \sin(\angle B)&=\dfrac{\blueD{AC}}{\goldD{AB}} \\\\ \sin(40^\circ)&=\dfrac{AC}{7}\quad\gray{\angle B=40^\circ, AB=7} \\\\ 7\cdot\sin(40^\circ)&=AC \end{aligned}
Now we evaluate using the calculator and round:
AC=7sin(40)4.5AC=7\cdot\sin(40^\circ)\approx 4.5
Problem 1.1
BC=BC=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Round your answer to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 2: Solving for an angle

Trigonometry can also be used to find missing angle measures. Let's find, for example, the measure of A\angle A in this triangle:
We are given the length of the side adjacent\purpleC{\text{adjacent}} to the missing angle, and the length of the hypotenuse\goldD{\text{hypotenuse}}. The trigonometric ratio that contains both of those sides is the cosine:
cos(A)=ACABcos(A)=68AC=6,AB=8A=cos1(68)\begin{aligned} \cos(\angle A)&=\dfrac{\purpleC{AC}}{\goldD{AB}} \\\\ \cos(\angle A)&=\dfrac{6}{8}\quad\gray{AC=6, AB=8} \\\\ \angle A&=\cos^{-1}\left(\dfrac{6}{8}\right) \end{aligned}
Now we evaluate using the calculator and round:
A=cos1(68)41.41\angle A=\cos^{-1}\left(\dfrac{6}{8}\right) \approx 41.41^\circ
Problem 2.1
A=\angle A=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
^\circ
Round your answer to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 3: Right triangle word problems

Problem 3.1
Howard is designing a chair swing ride. The swing ropes are 55 meters long, and in full swing they tilt in an angle of 2929^\circ. Howard wants the chairs to be 2.752.75 meters above the ground in full swing.
How tall should the pole of the swing ride be?
Round your final answer to the nearest hundredth.
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
meters

Want to try more problems like this? Check out this exercise.
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