**Example Trigonometric Problem for Educational Website**### Problem Statement:Find the exact value of the trigonometric expression given that:\[ \sin(u) = -\frac{3}{5}, \]where $ \frac{3\pi}{2} < u < 2\pi $, and\[ \cos(v) = \frac{15}{17}, \]where $ 0 < v < \frac{\pi}{2} $.### Expression to Find:\[ \sec(v - u) \]### Steps to Solve:1. **Find the Cosine of $ u $**: Given: $\sin(u) = -\frac{3}{5}$. Using the Pythagorean identity: \[ \cos^2(u) + \sin^2(u) = 1, \] \[ \cos^2(u) = 1 - \sin^2(u), \] \[ \cos^2(u) = 1 - \left(-\frac{3}{5}\right)^2, \] \[ \cos^2(u) = 1 - \frac{9}{25}, \] \[ \cos^2(u) = \frac{16}{25}, \] \[ \cos(u) = \pm \frac{4}{5}. \] Since $ \frac{3\pi}{2} < u < 2\pi $ (fourth quadrant), where cosine is positive: \[ \cos(u) = \frac{4}{5}. \]2. **Find the Sine of $ v $**: Given: $ \cos(v) = \frac{15}{17} $. Using the Pythagorean identity: \[ \sin^2(v) + \cos^2(v) = 1, \] \[ \sin^2(v) = 1 - \cos^2(v), \] \[ \sin^2(v) = 1 - \left(\frac{15}{17}\right)^2, \] \[ \sin^2(v) = 1 - \frac{225}{289}, \] \[ \sin^2(v) = \frac{64}{289}, \] \[ \sin(v) = \pm \frac{

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Description: **Example Trigonometric Problem for Educational Website**### Problem Statement:Find the exact value of the trigonometric expression given that:\[ \sin(u) = -\frac{3}{5}, \]where $ \frac{3\pi}{2} < u < 2\pi $, and\[ \cos(v) = \frac{15}{17}, \]where

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Find the exact value of the trigonometric expression given that sin(u) = - 3 where 371/2 < u < 27t, and cos(v) = 5' 15 where 0 < v < T/2. sec(v - u) Nood Holn?

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter2: Right Triangle Trigonometry

Section: Chapter Questions

Problem 1RP: The origins of the sine function are found in the tables of chords for a circle constructed by the...

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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**Example Trigonometric Problem for Educational Website**

### Problem Statement:

Find the exact value of the trigonometric expression given that:
\[ \sin(u) = -\frac{3}{5}, \]
where $ \frac{3\pi}{2} < u < 2\pi $, and
\[ \cos(v) = \frac{15}{17}, \]
where $ 0 < v < \frac{\pi}{2} $.

### Expression to Find:
\[ \sec(v - u) \]

### Steps to Solve:

1. **Find the Cosine of $ u $**:
Given: $\sin(u) = -\frac{3}{5}$.

Using the Pythagorean identity:
\[ \cos^2(u) + \sin^2(u) = 1, \]
\[ \cos^2(u) = 1 - \sin^2(u), \]
\[ \cos^2(u) = 1 - \left(-\frac{3}{5}\right)^2, \]
\[ \cos^2(u) = 1 - \frac{9}{25}, \]
\[ \cos^2(u) = \frac{16}{25}, \]
\[ \cos(u) = \pm \frac{4}{5}. \]

Since $ \frac{3\pi}{2} < u < 2\pi $ (fourth quadrant), where cosine is positive:
\[ \cos(u) = \frac{4}{5}. \]

2. **Find the Sine of $ v $**:
Given: $ \cos(v) = \frac{15}{17} $.

Using the Pythagorean identity:
\[ \sin^2(v) + \cos^2(v) = 1, \]
\[ \sin^2(v) = 1 - \cos^2(v), \]
\[ \sin^2(v) = 1 - \left(\frac{15}{17}\right)^2, \]
\[ \sin^2(v) = 1 - \frac{225}{289}, \]
\[ \sin^2(v) = \frac{64}{289}, \]
\[ \sin(v) = \pm \frac{

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