Question

suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (20/29, - 21/29). find the exact values of cotθ, cosθ, and cscθ. cotθ = cosθ = cscθ =

Explanation:

Step1: Recall trigonometric - function definitions

For a point $(x,y)$ on the unit - circle, $\cos\theta=x$, $\sin\theta = y$, $\cot\theta=\frac{\cos\theta}{\sin\theta}$, and $\csc\theta=\frac{1}{\sin\theta}$. Here, $x = \frac{20}{29}$ and $y=-\frac{21}{29}$.

Step2: Calculate $\cos\theta$

Since $\cos\theta=x$, then $\cos\theta=\frac{20}{29}$.

Step3: Calculate $\sin\theta$

Since $\sin\theta = y$, then $\sin\theta=-\frac{21}{29}$.

Step4: Calculate $\cot\theta$

Using the formula $\cot\theta=\frac{\cos\theta}{\sin\theta}$, substitute $\cos\theta=\frac{20}{29}$ and $\sin\theta = -\frac{21}{29}$, we get $\cot\theta=\frac{\frac{20}{29}}{-\frac{21}{29}}=-\frac{20}{21}$.

Step5: Calculate $\csc\theta$

Using the formula $\csc\theta=\frac{1}{\sin\theta}$, substitute $\sin\theta=-\frac{21}{29}$, we get $\csc\theta=-\frac{29}{21}$.

Answer:

$\cot\theta=-\frac{20}{21}$
$\cos\theta=\frac{20}{29}$
$\csc\theta=-\frac{29}{21}$