Write each of the following in terms of the cosine of another angle between 0° and 360°. a) tan(85)

To express \(\tan(85^\circ)\) in terms of the cosine of another angle between 0° and 360°, we can start by using the identity that relates the tangent function to the cosine function:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

Since there is no immediate simplification involving just cosine, we can use angle transformations to express \(\tan(85^\circ)\) in terms of another angle whose trigonometric function we can simplify. One such transformation is:

\[
\tan(85^\circ) = \cot(5^\circ)
\]

Now, we can use the fact that:

\[
\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}
\]

Thus, \(\tan(85^\circ)\) becomes:

\[
\tan(85^\circ) = \frac{\cos(5^\circ)}{\sin(5^\circ)}
\]

This expression involves the cosine function of the angle \(5^\circ\), fulfilling the requirement of expressing \(\tan(85^\circ)\) in terms of the cosine of another angle.

### Would you like further details or explanations on this?

Here are some related questions you could explore:
1. How would you express \(\tan(175^\circ)\) in terms of another angle?
2. What identity can be used to simplify \(\cot(90^\circ - \theta)\)?
3. Can \(\tan(85^\circ)\) be expressed purely in terms of \(\cos(85^\circ)\)?
4. How would you express \(\sin(85^\circ)\) in terms of a different angle?
5. How does \(\tan(90^\circ - \theta)\) relate to \(\cot(\theta)\)?

**Tip:** When simplifying trigonometric expressions, try using known angle identities like co-function identities, such as \(\sin(90^\circ - \theta) = \cos(\theta)\).

Learn how to express tan(85°) in terms of the cosine of another angle using trigonometric identities. This solution explains key trigonometric concepts such as the tangent and cotangent functions and their relation to cosine. Suitable for students in Grades 10-12.

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