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

Practice set 3: Right triangle word problems

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