$$ \text{if } $$$$ \tan\theta=-\frac13 $$and teta is in the fourth quarter then what is Sin and Cos of it

To find $ \sin \theta $ and $ \cos \theta $ given that $ \tan \theta = -\frac{1}{3} $ and $ \theta $ is in the fourth quadrant, we can proceed as follows:

### Step 1: Understanding the Signs of Sine and Cosine
Since $ \theta $ is in the fourth quadrant:
- $ \cos \theta $ is positive.
- $ \sin \theta $ is negative.

### Step 2: Using the Definition of Tangent
We know that:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{1}{3}
\]
Let’s assume:
\[
\sin \theta = -1 \quad \text{and} \quad \cos \theta = 3
\]
Then, we can construct a right triangle with these values.

### Step 3: Calculate the Hypotenuse
Using the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]

### Step 4: Finding $ \sin \theta $ and $ \cos \theta $
Now, we have:
\[
\sin \theta = \frac{-1}{\sqrt{10}} = -\frac{1}{\sqrt{10}}
\]
\[
\cos \theta = \frac{3}{\sqrt{10}} = \frac{3}{\sqrt{10}}
\]

To rationalize these:
\[
\sin \theta = -\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = -\frac{\sqrt{10}}{10}
\]
\[
\cos \theta = \frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{3\sqrt{10}}{10}
\]

### Final Answer
\[
\sin \theta = -\frac{\sqrt{10}}{10}
\]
\[
\cos \theta = \frac{3\sqrt{10}}{10}
\]

Would you like more details or have any questions?

### Related Questions
1. How do we determine the sign of trigonometric functions in each quadrant?
2. What is the exact value of $ \tan \theta $ in other quadrants?
3. How can we derive similar values if $ \tan \theta $ were positive?
4. How do we verify if our answer satisfies the Pythagorean identity $ \sin^2 \theta + \cos^2 \theta = 1 $?
5. Can these values be used to find $ \csc \theta $, $ \sec \theta $, and $ \cot \theta $?

### Tip
Always check the quadrant to determine the sign of trigonometric functions!

If $ \tan \theta = -\frac{1}{3} $ and $ \theta $ is in the fourth quadrant, then what is $ \sin \theta $ and $ \cos \theta $?

Discover how to determine $ \sin \theta $ and $ \cos \theta $ when $ \tan \theta = -\frac{1}{3} $ and $ \theta $ is in the fourth quadrant. This trigonometry problem involves understanding quadrant signs, using the Pythagorean theorem, and applying trigonometric identities. Ideal for students in Grades 10-12.

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