Question

use the figure to find the exact value of the following trigonometric function.

cos(2\alpha)

cos(2\alpha)=\square (simplify your answer.)

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 7\) and \(b=24\).
\(c=\sqrt{7^{2}+24^{2}}=\sqrt{49 + 576}=\sqrt{625}=25\)

Step2: Find \(\cos\alpha\)

By the definition of cosine in a right - triangle \(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos\alpha=\frac{7}{25}\)

Step3: Use the double - angle formula for cosine

The double - angle formula \(\cos(2\alpha)=2\cos^{2}\alpha - 1\)
Substitute \(\cos\alpha=\frac{7}{25}\) into the formula:

$$
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$$

Answer:

\(-\frac{527}{625}\)