sin theta = 0.42 explain how to find sin(pi+theta)

To find \(\sin(\pi + \theta)\) given \(\sin(\theta) = 0.42\), you need to use trigonometric identities. Let's walk through the process:

### Step 1: Use the Trigonometric Identity for \(\sin(\pi + \theta)\)

The identity for \(\sin(\pi + \theta)\) is:
\[
\sin(\pi + \theta) = -\sin(\theta)
\]
This identity comes from the symmetry properties of the sine function in different quadrants. Specifically:
- The angle \(\pi + \theta\) moves you into the third quadrant of the unit circle.
- In the third quadrant, sine is negative, and the sine of the angle \(\pi + \theta\) is the negative of the sine of the angle \(\theta\).

### Step 2: Apply the Identity

Given \(\sin(\theta) = 0.42\), we apply the identity:
\[
\sin(\pi + \theta) = -\sin(\theta)
\]
Thus:
\[
\sin(\pi + \theta) = -0.42
\]

### Explanation

The reason why the sine of \(\pi + \theta\) is negative is due to the way angles are measured on the unit circle. In the third quadrant (where angles are between \(\pi\) and \(3\pi/2\)), the sine function (which corresponds to the \(y\)-coordinate on the unit circle) is negative.

### Conclusion

Therefore, if \(\sin(\theta) = 0.42\), then:
\[
\sin(\pi + \theta) = -0.42
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the cosine of \(\pi + \theta\) given \(\sin(\theta) = 0.42\)?
2. How does the unit circle help in understanding trigonometric functions in different quadrants?
3. Can you derive the formula for \(\cos(\pi + \theta)\)?
4. What are the signs of sine and cosine in all four quadrants?
5. How do trigonometric identities simplify the computation of angles?

**Tip:** Remember that symmetry and signs of trigonometric functions depend on the quadrant in which the angle lies.

Explore how to find \( \sin(\pi + \theta) \) when given \( \sin(\theta) = 0.42 \). Learn about trigonometric identities and the unit circle's role in determining sine values in different quadrants. Suitable for high school students studying trigonometry.

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