Questions: Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (15/17, -8/17). Find the exact values of sin θ, sec θ, and cot θ. sin θ= sec θ= cot θ=

Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (15/17, -8/17).
Find the exact values of sin θ, sec θ, and cot θ.

sin θ=
sec θ=
cot θ=

Transcript text: Suppose that $\theta$ is an angle in standard position whose terminal side intersects the unit circle at $\left(\frac{15}{17},-\frac{8}{17}\right)$. Find the exact values of $\sin \theta, \sec \theta$, and $\cot \theta$. \[ \begin{array}{l} \sin \theta=\square \\ \sec \theta=\square \\ \cot \theta=\square \end{array} \]

Solution

Solution Steps

Step 1: Calculate $\sin \theta$

The sine of the angle $\theta$ is given by the y-coordinate of the point where the terminal side intersects the unit circle. Thus, we have: \[ \sin \theta = -\frac{8}{17} \]

Step 2: Calculate $\sec \theta$

The secant of the angle $\theta$ is the reciprocal of the x-coordinate of the intersection point. Therefore, we find: \[ \sec \theta = \frac{1}{\frac{15}{17}} = \frac{17}{15} \approx 1.1333 \]

Step 3: Calculate $\cot \theta$

The cotangent of the angle $\theta$ is the ratio of the x-coordinate to the y-coordinate. Thus, we compute: \[ \cot \theta = \frac{\frac{15}{17}}{-\frac{8}{17}} = -\frac{15}{8} \approx -1.875 \]

Final Answer

\[ \sin \theta = -\frac{8}{17}, \quad \sec \theta \approx 1.1333, \quad \cot \theta \approx -1.875 \] Thus, the final boxed answers are: \[ \boxed{\sin \theta = -\frac{8}{17}, \sec \theta \approx 1.1333, \cot \theta \approx -1.875} \]

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