use the adjacent figure to find the exact value of the following trigonometric function. $\\cos \\frac {…

Explanation:

Step1: Find the hypotenuse

By Pythagorean theorem (c=\sqrt{a^{2}+b^{2}}), where (a = 3), (b=4). (c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5) So, (\cos\alpha=\frac{3}{5})

Step2: Use the half - angle formula

The half - angle formula for cosine is (\cos\frac{\theta}{2}=\sqrt{\frac{1+\cos\theta}{2}}) (since (\alpha) is an acute angle in a right - triangle, (\frac{\alpha}{2}) is also acute and (\cos\frac{\alpha}{2}>0)) Substitute (\theta=\alpha) and (\cos\alpha=\frac{3}{5}) into the formula: (\cos\frac{\alpha}{2}=\sqrt{\frac{1+\frac{3}{5}}{2}}=\sqrt{\frac{\frac{5 + 3}{5}}{2}}=\sqrt{\frac{\frac{8}{5}}{2}}=\sqrt{\frac{4}{5}}=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5})

Answer:

(\frac{2\sqrt{5}}{5})