Let 𝛼
 be an angle, with 0≤𝛼<2𝜋
. Given cos(2𝛼)=−4781
 and 𝜋/2<2𝛼<𝜋
, find exact values of the six trigonometric functions.
Note: You are not allowed to use decimals in your answer.

sin(𝛼)
 = 
 .

cos(𝛼)
 = 
 .

tan(𝛼)
 = 
 .

csc(𝛼)
 = 
 .

sec(𝛼)
 = 
 .

cot(𝛼)
 =

We are given that \( \cos(2\alpha) = -4781 \) (which seems very unusual because the cosine of an angle cannot exceed 1 or be less than -1) and that \( \frac{\pi}{2} < 2\alpha < \pi \). Let’s assume this was a typo, and the more reasonable scenario might be that \( \cos(2\alpha) = -\frac{4}{5} \), which is within the valid range for cosine and provides a practical problem. Let’s proceed with this assumption and solve it:

### Step 1: Use \( \cos(2\alpha) = 2\cos^2(\alpha) - 1 \) to find \( \cos(\alpha) \)

We know the double angle identity for cosine:

\[
\cos(2\alpha) = 2\cos^2(\alpha) - 1
\]

Substitute \( \cos(2\alpha) = -\frac{4}{5} \) into the equation:

\[
-\frac{4}{5} = 2\cos^2(\alpha) - 1
\]

Solve for \( \cos^2(\alpha) \):

\[
2\cos^2(\alpha) = -\frac{4}{5} + 1 = \frac{1}{5}
\]

\[
\cos^2(\alpha) = \frac{1}{10}
\]

Thus,

\[
\cos(\alpha) = \pm \frac{1}{\sqrt{10}}
\]

### Step 2: Determine the sign of \( \cos(\alpha) \)

Since \( \frac{\pi}{2} < 2\alpha < \pi \), the angle \( \alpha \) lies in the second quadrant. In the second quadrant, cosine is negative, so:

\[
\cos(\alpha) = -\frac{1}{\sqrt{10}}
\]

### Step 3: Find \( \sin(\alpha) \)

Use the Pythagorean identity \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \) to find \( \sin(\alpha) \):

\[
\sin^2(\alpha) = 1 - \cos^2(\alpha) = 1 - \frac{1}{10} = \frac{9}{10}
\]

Thus,

\[
\sin(\alpha) = \pm \frac{3}{\sqrt{10}}
\]

Since \( \alpha \) is in the second quadrant, sine is positive, so:

\[
\sin(\alpha) = \frac{3}{\sqrt{10}}
\]

### Step 4: Find the remaining trigonometric functions

- **Tangent:** 
  \[
  \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{\frac{3}{\sqrt{10}}}{-\frac{1}{\sqrt{10}}} = -3
  \]

- **Cosecant:**
  \[
  \csc(\alpha) = \frac{1}{\sin(\alpha)} = \frac{\sqrt{10}}{3}
  \]

- **Secant:**
  \[
  \sec(\alpha) = \frac{1}{\cos(\alpha)} = -\sqrt{10}
  \]

- **Cotangent:**
  \[
  \cot(\alpha) = \frac{1}{\tan(\alpha)} = -\frac{1}{3}
  \]

### Final Answer:

- \( \sin(\alpha) = \frac{3}{\sqrt{10}} \)
- \( \cos(\alpha) = -\frac{1}{\sqrt{10}} \)
- \( \tan(\alpha) = -3 \)
- \( \csc(\alpha) = \frac{\sqrt{10}}{3} \)
- \( \sec(\alpha) = -\sqrt{10} \)
- \( \cot(\alpha) = -\frac{1}{3} \)

Would you like more details or have any questions?

### Follow-up questions:
1. How would the solution change if \( \alpha \) were in the first quadrant?
2. What is the geometric interpretation of the given trigonometric values?
3. Can you solve for \( \alpha \) exactly based on these values?
4. How would you find the values of the trigonometric functions for \( 2\alpha \) directly?
5. What would happen if \( \cos(2\alpha) \) had a different sign?

### Tip: 
Always remember which quadrant your angle is in, as it determines the signs of sine, cosine, and tangent.

Let 𝛼 be an angle, with 0 ≤ 𝛼 < 2𝜋. Given cos(2𝛼) = -4781 and 𝜋/2 < 2𝛼 < 𝜋, find exact values of the six trigonometric functions. Note: You are not allowed to use decimals in your answer.

sin(𝛼) = .

cos(𝛼) = .

tan(𝛼) = .

csc(𝛼) = .

sec(𝛼) = .

cot(𝛼) = .

Learn how to find exact values for sin(𝛼), cos(𝛼), tan(𝛼), csc(𝛼), sec(𝛼), and cot(𝛼) given cos(2𝛼) and the range of 𝛼. This problem involves double angle identities and Pythagorean identities in trigonometry. Suitable for Grades 10-12.

Math Problem Statement

sin(𝛼)

cos(𝛼)

tan(𝛼)

csc(𝛼)

sec(𝛼)

cot(𝛼)

Solution

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