Question

find the exact values of the six trigonometric ratios of the angle θ in the triangle. sin(θ) = cos(θ)= tan(θ)= csc(θ)= sec(θ)= cot(θ)=

Explanation:

Step1: Find the adjacent side

Let the hypotenuse $c = 9$, the opposite - side $a = 8$. Using the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, we get $b=\sqrt{c^{2}-a^{2}}=\sqrt{9^{2}-8^{2}}=\sqrt{81 - 64}=\sqrt{17}$.

Step2: Calculate $\sin(\theta)$

By the definition of sine, $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{9}$.

Step3: Calculate $\cos(\theta)$

By the definition of cosine, $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{\sqrt{17}}{9}$.

Step4: Calculate $\tan(\theta)$

By the definition of tangent, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{\sqrt{17}}=\frac{8\sqrt{17}}{17}$.

Step5: Calculate $\csc(\theta)$

Since $\csc(\theta)=\frac{1}{\sin(\theta)}$, then $\csc(\theta)=\frac{9}{8}$.

Step6: Calculate $\sec(\theta)$

Since $\sec(\theta)=\frac{1}{\cos(\theta)}$, then $\sec(\theta)=\frac{9}{\sqrt{17}}=\frac{9\sqrt{17}}{17}$.

Step7: Calculate $\cot(\theta)$

Since $\cot(\theta)=\frac{1}{\tan(\theta)}$, then $\cot(\theta)=\frac{\sqrt{17}}{8}$.

Answer:

$\sin(\theta)=\frac{8}{9}$
$\cos(\theta)=\frac{\sqrt{17}}{9}$
$\tan(\theta)=\frac{8\sqrt{17}}{17}$
$\csc(\theta)=\frac{9}{8}$
$\sec(\theta)=\frac{9\sqrt{17}}{17}$
$\cot(\theta)=\frac{\sqrt{17}}{8}$