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. a = √2, c = 2

Accepted Answer

Identify the sides of the right triangle ABC, where the right angle is at C. This means side c is the hypotenuse, and sides a and b are the legs opposite angles A and B respectively. Use the Pythagorean theorem, which states $$a^2$$ + $$b^2$$ = $$c^2$$, to find the unknown side length $$b. $$Substitute the known values $$a = \sqrt{2}$$ and $$c = 2$$ into the equation: $$\left(\sqrt{2}\$$right)^2$$ + b^2$ = 2^2$. Simplify the equation to solve for b^2$: 2$$ + $$b^2 = 4$, then isolate b^2$ by subtracting 2 from both sides to get b^2$ = 2. $$Finally, take the positive square root to find $$b = \sqrt{2}$$, since side lengths are positive. Now, find the six trigonometric functions for angle B. Recall the definitions relative to angle B: $$\sin B = \frac{	ext{opposite}}{	ext{hypotenuse}} = \frac{a}{c}$$, $$\cos B = \frac{	ext{adjacent}}{	ext{hypotenuse}} = \frac{b}{c}$$, and $$	an B = \frac{	ext{opposite}}{	ext{adjacent}} = \frac{a}{b}. $$Use the values $$a = \sqrt{2}$$, $$b = \sqrt{2}$$, and $$c = 2 to $$write these ratios. Next, find the reciprocal functions: $$\csc B = \frac{1}{\sin B}$$, $$\sec B = \frac{1}{\cos B}$$, and $$\cot B = \frac{1}{	an B}. $$Simplify each expression and rationalize denominators where necessary to express the exact values.

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. a = √2, c = 2

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