Suppose that  is an angle in the third quadrant, and sin = − 4 . Find cos and tan .

Given that \(\theta\) is an angle in the third quadrant and \(\sin \theta = -\frac{4}{5}\), we need to find \(\cos \theta\) and \(\tan \theta\).

### Step 1: Use the Pythagorean identity
The Pythagorean identity is:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Substitute \(\sin \theta = -\frac{4}{5}\):
\[
\left(-\frac{4}{5}\right)^2 + \cos^2 \theta = 1
\]
\[
\frac{16}{25} + \cos^2 \theta = 1
\]

### Step 2: Solve for \(\cos^2 \theta\)
Subtract \(\frac{16}{25}\) from both sides:
\[
\cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
\]

### Step 3: Determine \(\cos \theta\)
Take the square root of both sides:
\[
\cos \theta = \pm \frac{3}{5}
\]

Since \(\theta\) is in the third quadrant, both sine and cosine are negative:
\[
\cos \theta = -\frac{3}{5}
\]

### Step 4: Calculate \(\tan \theta\)
Now, use the identity:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
\[
\tan \theta = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}
\]

### Final Answers
- \(\cos \theta = -\frac{3}{5}\)
- \(\tan \theta = \frac{4}{3}\)

Would you like further details or have any other questions?

Here are some related questions:
1. What is the value of \(\cot \theta\) in the third quadrant?
2. How do you find the reference angle for \(\theta\) in the third quadrant?
3. What are the signs of trigonometric functions in each quadrant?
4. How can you find \(\sec \theta\) and \(\csc \theta\) from \(\sin \theta\) and \(\cos \theta\)?
5. What is the significance of the Pythagorean identity in trigonometry?

**Tip:** Always remember that in the third quadrant, sine and cosine are both negative, but tangent is positive.

Discover how to find cosine and tangent in the third quadrant when sine is -4/5. This trigonometry problem utilizes the Pythagorean identity and tangent definition to derive the solutions step-by-step. Ideal for high school students studying trigonometric functions.

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