Use the fundamental identities to find the value of the trigonometric function.

Find

tangent thetatanθ​,

given that

secant theta equals nine halvessecθ=92

and

thetaθ

is in quadrant IV.

To find \( \tan \theta \) given \( \sec \theta = \frac{9}{2} \) and \( \theta \) is in Quadrant IV, we'll use trigonometric identities and the fact that in Quadrant IV, \( \tan \theta \) is negative. Here's the step-by-step process:

### Step 1: Recall the relationship between \( \sec \theta \) and \( \cos \theta \)
The secant function is the reciprocal of the cosine function, so:
\[
\sec \theta = \frac{1}{\cos \theta}
\]
Given that \( \sec \theta = \frac{9}{2} \), we have:
\[
\cos \theta = \frac{2}{9}
\]

### Step 2: Use the Pythagorean identity to find \( \sin \theta \)
The Pythagorean identity states:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \( \cos \theta = \frac{2}{9} \) into the identity:
\[
\sin^2 \theta + \left( \frac{2}{9} \right)^2 = 1
\]
\[
\sin^2 \theta + \frac{4}{81} = 1
\]
Now, subtract \( \frac{4}{81} \) from both sides:
\[
\sin^2 \theta = 1 - \frac{4}{81} = \frac{81}{81} - \frac{4}{81} = \frac{77}{81}
\]
So,
\[
\sin \theta = \pm \sqrt{\frac{77}{81}} = \pm \frac{\sqrt{77}}{9}
\]

### Step 3: Determine the sign of \( \sin \theta \)
Since \( \theta \) is in Quadrant IV, and the sine function is negative in this quadrant, we have:
\[
\sin \theta = -\frac{\sqrt{77}}{9}
\]

### Step 4: Use the definition of tangent to find \( \tan \theta \)
We know that:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
Substitute the values of \( \sin \theta \) and \( \cos \theta \):
\[
\tan \theta = \frac{-\frac{\sqrt{77}}{9}}{\frac{2}{9}} = -\frac{\sqrt{77}}{2}
\]

Thus, the value of \( \tan \theta \) is:
\[
\tan \theta = -\frac{\sqrt{77}}{2}
\]

### Would you like more details or have any questions?

Here are 5 related questions to expand on this problem:

1. How would the value of \( \tan \theta \) change if \( \theta \) were in Quadrant I instead of Quadrant IV?
2. Can you find \( \sin \theta \) if you are given \( \csc \theta \) and the quadrant?
3. What is the process for finding \( \cot \theta \) from \( \sec \theta \)?
4. How does the sign of \( \cos \theta \) change in different quadrants?
5. How can we use trigonometric identities to find other functions given \( \tan \theta \)?

**Tip:** Always pay attention to the quadrant when solving trigonometric problems, as it determines the signs of the trigonometric functions.

Use the fundamental identities to find the value of the trigonometric function. Find tangent θ (tan θ), given that secant θ equals nine halves (sec θ = 9/2) and θ is in quadrant IV.

Discover how to use fundamental trigonometric identities to find tan θ when sec θ = 9/2 and θ is in Quadrant IV. This problem involves applying the Pythagorean identity and understanding the behavior of trigonometric functions in different quadrants. Suitable for high school students in Grades 10-12.

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