Find the exact value of s in the given interval that has the given circular function value.

Do not use a calculator.

left bracket StartFraction pi Over 2 EndFraction comma pi right bracketπ2,π​;

tangent stansequals=negative StartRoot 3 EndRoot−3

Question content area bottom

Part 1

sequals=enter your response here

radians

​(Simplify your answer. Type an exact​ answer, using

piπ

as needed. Use integers or fractions for any numbers in the​ expression.)

We are tasked with finding the value of \( s \) in the interval \( \left[ \frac{\pi}{2}, \pi \right] \) such that:

\[
\tan(s) = -\sqrt{3}
\]

### Step 1: Determine the reference angle
We know that \( \tan(s) = -\sqrt{3} \). The tangent function of \( \frac{\pi}{3} \) is \( \sqrt{3} \), so the reference angle is:

\[
\theta = \frac{\pi}{3}
\]

### Step 2: Find the quadrant
The interval \( \left[ \frac{\pi}{2}, \pi \right] \) corresponds to the second quadrant. In the second quadrant, the tangent function is negative, which fits with \( \tan(s) = -\sqrt{3} \).

### Step 3: Calculate \( s \)
In the second quadrant, the value of \( s \) will be:

\[
s = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3}
\]

Thus, the exact value of \( s \) is:

\[
s = \frac{2\pi}{3}
\]

### Final Answer:
\[
s = \frac{2\pi}{3} \text{ radians}
\]

Would you like more details or have any questions about this solution?

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Here are 5 related questions to consider:
1. What is the value of \( \tan\left( \frac{\pi}{3} \right) \)?
2. How does the tangent function behave in different quadrants?
3. What is the reference angle for \( \tan^{-1}(-\sqrt{3}) \)?
4. Why is the tangent function negative in the second quadrant?
5. What other angles have the same tangent value as \( s \) in other intervals?

**Tip:** Always remember that tangent is positive in the first and third quadrants, and negative in the second and fourth.

Find the exact value of s in the given interval that has the given circular function value. [π/2, π]; tan(s) = -√3. Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.

Solve the trigonometric problem: Find the exact value of s in the interval [π/2, π] where tan(s) = -√3. Learn about tangent functions, reference angles, and their behavior in different quadrants with a detailed step-by-step solution. Suitable for high school students in Grades 10-12.

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