Question

assume that θ is an acute angle in a right - triangle satisfying the given condition. evaluate the remaining trigonometric functions. cosθ = 5/13 tanθ = 12/5 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) cscθ = 13/12 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) secθ = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of cosine

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\cos\theta = \frac{5}{13}$, let the adjacent side to the angle $\theta$ be $a = 5$ and the hypotenuse $c=13$.

Step2: Use the Pythagorean theorem to find the opposite side

By the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, where $b$ is the opposite side to the angle $\theta$. So $b=\sqrt{c^{2}-a^{2}}=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$.

Step3: Recall the definition of secant

The secant function is defined as $\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}$.
Since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{5}{13}$, then $\sec\theta=\frac{1}{\cos\theta}=\frac{13}{5}$.

Answer:

$\frac{13}{5}$