Find a) cos520° b) cos(-200°) c) cos(-160°) d) cos200° e) cos240° f) cos20° g) sin250° h) sin70° when cos160° = -0.940

Let's work through each of these cosine and sine values, using known trigonometric identities and properties of angles.

### a) \( \cos 520^\circ \)

Since the cosine function is periodic with a period of \(360^\circ\), we reduce the angle:
\[
520^\circ - 360^\circ = 160^\circ
\]
Thus, \( \cos 520^\circ = \cos 160^\circ = -0.940 \).

### b) \( \cos(-200^\circ) \)

Using the identity \( \cos(-\theta) = \cos(\theta) \):
\[
\cos(-200^\circ) = \cos(200^\circ)
\]
We'll find \( \cos 200^\circ \) in part **d**.

### c) \( \cos(-160^\circ) \)

Similarly, using \( \cos(-\theta) = \cos(\theta) \):
\[
\cos(-160^\circ) = \cos(160^\circ) = -0.940
\]

### d) \( \cos 200^\circ \)

First, note that \( 200^\circ \) is in the third quadrant where cosine is negative. We can subtract \( 200^\circ \) from \( 180^\circ \) to find the reference angle:
\[
200^\circ - 180^\circ = 20^\circ
\]
Thus:
\[
\cos 200^\circ = -\cos 20^\circ
\]
Since we haven't calculated \( \cos 20^\circ \) yet, we'll find that in part **f**.

### e) \( \cos 240^\circ \)

\( 240^\circ \) is also in the third quadrant. The reference angle is:
\[
240^\circ - 180^\circ = 60^\circ
\]
Thus:
\[
\cos 240^\circ = -\cos 60^\circ = -\frac{1}{2}
\]

### f) \( \cos 20^\circ \)

Using a calculator, we find:
\[
\cos 20^\circ \approx 0.9397
\]

### g) \( \sin 250^\circ \)

\( 250^\circ \) is in the third quadrant where sine is negative. The reference angle is:
\[
250^\circ - 180^\circ = 70^\circ
\]
Thus:
\[
\sin 250^\circ = -\sin 70^\circ
\]
We'll calculate \( \sin 70^\circ \) in part **h**.

### h) \( \sin 70^\circ \)

Using a calculator:
\[
\sin 70^\circ \approx 0.9397
\]

Thus:
\[
\sin 250^\circ = -0.9397
\]

### Summary of Results:
a) \( \cos 520^\circ = -0.940 \)  
b) \( \cos(-200^\circ) = -\cos 20^\circ = -0.9397 \)  
c) \( \cos(-160^\circ) = -0.940 \)  
d) \( \cos 200^\circ = -0.9397 \)  
e) \( \cos 240^\circ = -\frac{1}{2} \)  
f) \( \cos 20^\circ \approx 0.9397 \)  
g) \( \sin 250^\circ = -0.9397 \)  
h) \( \sin 70^\circ \approx 0.9397 \)

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Do you need more details or have any questions?

Here are 5 related questions to expand on this:
1. What is the period of sine and cosine functions?
2. How does the unit circle help in calculating sine and cosine of various angles?
3. What is the relationship between sine and cosine for complementary angles?
4. How do sine and cosine change in different quadrants?
5. How can we derive sine and cosine values using trigonometric identities?

**Tip:** Remember that sine and cosine functions repeat every \(360^\circ\), so subtract or add multiples of \(360^\circ\) to simplify large or negative angles.

Explore the solutions to trigonometric functions for angles such as cos520°, cos(-200°), and sin250°. This detailed explanation covers cosine and sine periodicity, reference angles, and how these concepts apply in different quadrants. Perfect for students in Grades 10-12.

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