if sin x degrees is equal to 0.7 explain how to find the values of sin(180 degrees + x) degrees 

To find \(\sin(180^\circ + x^\circ)\) given that \(\sin(x^\circ) = 0.7\), we can apply a similar process using trigonometric identities and the properties of the sine function. Here's the step-by-step explanation:

### Step 1: Use the Trigonometric Identity for \(\sin(180^\circ + x^\circ)\)

The identity for \(\sin(180^\circ + x^\circ)\) is:
\[
\sin(180^\circ + x^\circ) = -\sin(x^\circ)
\]
This identity arises from the way angles are measured on the unit circle and the behavior of the sine function in different quadrants.

- The angle \(180^\circ + x^\circ\) places you in the third quadrant (where angles range from \(180^\circ\) to \(270^\circ\)).
- In the third quadrant, sine is negative.

### Step 2: Apply the Identity

Given that \(\sin(x^\circ) = 0.7\), we apply the identity:
\[
\sin(180^\circ + x^\circ) = -\sin(x^\circ)
\]
Thus:
\[
\sin(180^\circ + x^\circ) = -0.7
\]

### Explanation

The sine function is positive in the first and second quadrants but negative in the third and fourth quadrants. Since \(180^\circ + x^\circ\) places the angle in the third quadrant, the sine of the angle becomes the negative of the sine of the corresponding angle in the first quadrant. This is why \(\sin(180^\circ + x^\circ) = -0.7\).

### Conclusion

Therefore, if \(\sin(x^\circ) = 0.7\), then:
\[
\sin(180^\circ + x^\circ) = -0.7
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the cosine of \(\sin(180^\circ + x^\circ)\) if \(\sin(x^\circ) = 0.7\)?
2. How does the unit circle help in determining the signs of trigonometric functions in different quadrants?
3. Can you find the general solution for \(\sin(x^\circ) = 0.7\)?
4. How do trigonometric functions change as the angle moves across quadrants?
5. How can you apply the sine and cosine rule for non-right triangles?

**Tip:** When dealing with sine and cosine of angles in different quadrants, always remember to check the sign based on the quadrant's properties.

Discover how to find \(\sin(180^\circ + x^\circ)\) when given \(\sin(x^\circ) = 0.7\). Learn the step-by-step process using trigonometric identities and the properties of the sine function. Suitable for advanced high school students and anyone interested in trigonometry.

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