Determine the exact value of the following expression.
 
cos465°

To determine the exact value of \(\cos 465^\circ\), we can reduce the angle to a standard position within the range of \(0^\circ\) to \(360^\circ\) by subtracting \(360^\circ\) from \(465^\circ\):

\[
465^\circ - 360^\circ = 105^\circ
\]

Thus, \(\cos 465^\circ = \cos 105^\circ\).

Now, we need to determine the value of \(\cos 105^\circ\). Notice that \(105^\circ\) is in the second quadrant, where the cosine function is negative. Additionally, we can express \(105^\circ\) as:

\[
105^\circ = 180^\circ - 75^\circ
\]

Using the identity for cosine in the second quadrant, \(\cos(180^\circ - \theta) = -\cos \theta\), we get:

\[
\cos 105^\circ = -\cos 75^\circ
\]

The cosine of \(75^\circ\) can be calculated using the angle sum identity:

\[
\cos(75^\circ) = \cos(45^\circ + 30^\circ)
\]

Using the formula for the cosine of a sum:

\[
\cos(a + b) = \cos a \cos b - \sin a \sin b
\]

Substitute \(a = 45^\circ\) and \(b = 30^\circ\):

\[
\cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ
\]

Using the known values:

\[
\cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2}
\]

Substituting these into the equation:

\[
\cos 75^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)
\]

\[
\cos 75^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

Therefore:

\[
\cos 105^\circ = -\frac{\sqrt{6} - \sqrt{2}}{4}
\]

Thus, the exact value of \(\cos 465^\circ\) is:

\[
\cos 465^\circ = -\frac{\sqrt{6} - \sqrt{2}}{4}
\]

Would you like more details on any step or have any questions?

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Here are some related questions:

1. How do you find exact trigonometric values of other angles outside \(0^\circ\) to \(360^\circ\)?
2. What is the sine of \(465^\circ\)?
3. How does the cosine function behave in different quadrants?
4. Can you use the angle difference identity for similar problems?
5. What are some other trigonometric identities involving cosine?

**Tip**: Remember that trigonometric functions repeat their values every \(360^\circ\), which is called periodicity.

Determine the exact value of the following expression: cos 465°

Discover how to find the exact value of cos 465° by reducing the angle to within 0°-360°, using trigonometric identities and angle sum formulas. This step-by-step solution covers cosine in different quadrants and the periodicity of trigonometric functions, suitable for Grades 10-12.

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