Question

find the six trigonometric function values of the specified angle. sin θ = 7√74 / 74 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) cos θ = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the hypotenuse

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 5\) and \(b=7\). So \(c=\sqrt{5^{2}+7^{2}}=\sqrt{25 + 49}=\sqrt{74}\).

Step2: Calculate cosine

The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the adjacent side to \(\theta\) is \(5\) and the hypotenuse is \(\sqrt{74}\). So \(\cos\theta=\frac{5}{\sqrt{74}}=\frac{5\sqrt{74}}{74}\).

Answer:

\(\frac{5\sqrt{74}}{74}\)