Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11

Accepted Answer

Identify the sides of the right triangle ABC, where the right angle is at C. This means sides a and b are the legs, and side c is the hypotenuse opposite the right angle. Use the Pythagorean theorem, which states $$a^2$$ + $$b^2$$ = $$c^2$$, to find the unknown side length $$a. $$Substitute the known values $$b = 8$$ and $$c = 11$$ into the equation: $$a^2$$ + $$8^2$$ = $$11^2. $$Solve for $$a^2 by $$calculating $$a^2$$ = $$11^2$$ - $$8^2. $$Then take the positive square root to find $$a$$, since side lengths are positive. Once you have $$a$$, find the six trigonometric functions for angle $$B. $$Recall that angle $$B is $$adjacent to side $$b$$ and opposite side $$a$$, with hypotenuse $$c. $$The functions are: - $$\sin B = \frac{	ext{opposite}}{	ext{hypotenuse}} = \frac{a}{c}$$ - $$\cos B = \frac{	ext{adjacent}}{	ext{hypotenuse}} = \frac{b}{c}$$ - $$	an B = \frac{	ext{opposite}}{	ext{adjacent}} = \frac{a}{b}$$ - $$\csc B = \frac{1}{\sin B} = \frac{c}{a}$$ - $$\sec B = \frac{1}{\cos B} = \frac{c}{b}$$ - $$\cot B = \frac{1}{	an B} = \frac{b}{a}$$ Express each function in simplest exact form and rationalize denominators where necessary.

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11

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Concetti chiave

Pythagorean Theorem

Trigonometric Functions

Rationalizing Denominators