tan x = √ 3, cos x = - 1/2 find all trigonometric functions

Given:
\[
\tan x = \sqrt{3}, \quad \cos x = -\frac{1}{2}
\]

### Step 1: Determine \(x\) using \(\cos x = -\frac{1}{2}\)
From the cosine value, we know that \( \cos x = -\frac{1}{2} \). The cosine function equals \(-\frac{1}{2}\) at angles in the second and third quadrants:
- In the second quadrant: \( x = \frac{2\pi}{3} \)
- In the third quadrant: \( x = \frac{4\pi}{3} \)

### Step 2: Confirm with \(\tan x = \sqrt{3}\)
For \( \tan x = \sqrt{3} \), we know that this happens at:
- \( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \)
Thus, the angles where tangent is positive (since \(\sqrt{3}\) is positive) are in the first and third quadrants. Therefore, the correct solution in our case is the third quadrant, confirming:
\[
x = \frac{4\pi}{3}
\]

### Step 3: Find all trigonometric functions
Now that we know \(x = \frac{4\pi}{3}\), we can compute all the trigonometric functions for this angle:

1. **Sine:**
   \[
   \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}
   \]
   (Sine is negative in the third quadrant.)

2. **Cosine:**
   \[
   \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}
   \]
   (Cosine is negative in the third quadrant.)

3. **Tangent:**
   \[
   \tan\left(\frac{4\pi}{3}\right) = \frac{\sin x}{\cos x} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}
   \]
   (Tangent is positive in the third quadrant.)

4. **Cosecant:**
   \[
   \csc\left(\frac{4\pi}{3}\right) = \frac{1}{\sin x} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
   \]

5. **Secant:**
   \[
   \sec\left(\frac{4\pi}{3}\right) = \frac{1}{\cos x} = \frac{1}{-\frac{1}{2}} = -2
   \]

6. **Cotangent:**
   \[
   \cot\left(\frac{4\pi}{3}\right) = \frac{1}{\tan x} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
   \]

### Summary of Trigonometric Values:
- \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \)
- \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \)
- \( \tan\left(\frac{4\pi}{3}\right) = \sqrt{3} \)
- \( \csc\left(\frac{4\pi}{3}\right) = -\frac{2\sqrt{3}}{3} \)
- \( \sec\left(\frac{4\pi}{3}\right) = -2 \)
- \( \cot\left(\frac{4\pi}{3}\right) = \frac{\sqrt{3}}{3} \)

Would you like more details on any of the steps or functions?

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Here are 5 related questions to expand on this topic:
1. How do you find the reference angle for trigonometric functions?
2. What are the values of all trigonometric functions for \(\frac{2\pi}{3}\)?
3. How can you derive the tangent function from sine and cosine?
4. What is the periodicity of each trigonometric function?
5. How do you solve trigonometric equations involving multiple angles?

**Tip:** Always check the quadrant to determine the sign of the trigonometric function!

tan x = √3, cos x = -1/2 find all trigonometric functions

Discover how to find all the trigonometric functions for the equation tan x = √3 and cos x = -1/2. The solution covers key trigonometric concepts including sine, cosine, tangent, secant, cosecant, and cotangent, along with their signs in different quadrants. Ideal for students in Grades 10-12.

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