91原片

Solving for a Side Using Trig Ratios

*Given an angle and a given side, one can solve for a missing side using trig ratios.

  • Example 1: What is the length of the indicated side of the following triangle? The measure of angle A is 39掳, and the length of side b is 65 cm.
The image depicts a right triangle labeled with vertices A, B, and C. The triangle is filled with a blue color and oriented so that side BC is horizontal at the bottom, with point B on the left and point C on the right. The vertical side AB is indicated with the label "c," and the hypotenuse AC is labeled with the letter "b." The side BC, which is the base of the triangle, is marked with the letter "a." An arrow points from the vertex C towards the side labeled 鈥渁.鈥� Below the triangle, there is a question asking, 鈥淲hat is the length of this side?鈥� in black text.

The sides involved in this calculation are the opposite side (from angle A) and the hypotenuse. The trig ratio that relates the opposite side and hypotenuse is sine (opposite/hypotenuse).

Here is how the problem is set up: 聽聽聽聽 sin 39 = opposite side length hypotenuse length 听=听 a 65

By consulting a trig ratio table or using a calculator, one can see that sin 39 挺 = 0.963795.

Substituting that value in for sin 39 gives the following equation:

0.963795 = a 65

Solving for the variable a yields an answer of 63 cm.

Note: restrict the trig ratios you are using to sine, cosine, and tangent.

  • Example 2: Using the same triangle from the previous problem, find the length of side c given that angle A is 39 挺 and the length of side a is 63 cm (the length of the hypotenuse is the same as the previous problem 鈥� 65 cm).

Side c is the adjacent side to angle A, so the trig ratio used will be cosine. Here is how the problem will be set up:

cos 39 = adjacent side length hypotenuse 听=听 c 65

By consulting a trig ratio table or by using a calculator to determine the cos 39 挺 one can see that the cos 39 掳 = 0.266643

Substituting that value in for cos 39 挺 gives the following equation:

0.266643 = c 65

Solving for the variable c yields an answer of 17 cm.