Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1.
cos(105°)

To find the exact value of \( \cos(105^\circ) \), we can use the **cosine addition formula**. We know that:

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

Since \( 105^\circ = 60^\circ + 45^\circ \), we can apply the formula as follows:

\[
\cos(105^\circ) = \cos(60^\circ + 45^\circ)
\]

### Step-by-Step Solution:

1. **Apply the Cosine Addition Formula:**
   \[
   \cos(105^\circ) = \cos(60^\circ) \cos(45^\circ) - \sin(60^\circ) \sin(45^\circ)
   \]

2. **Substitute Known Values:**
   - \( \cos(60^\circ) = \frac{1}{2} \)
   - \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \)
   - \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
   - \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)

   Substituting these values gives:
   \[
   \cos(105^\circ) = \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) - \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right)
   \]

3. **Simplify the Expression:**
   \[
   \cos(105^\circ) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}
   \]

4. **Combine Terms:**
   \[
   \cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4}
   \]

### Final Answer:
\[
\cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4}
\]

Would you like further details on the steps?

---

Here are five related questions to deepen your understanding:

1. How would you use the sine addition formula to find \( \sin(105^\circ) \)?
2. What is the exact value of \( \cos(75^\circ) \) using the same approach?
3. How can the cosine subtraction formula be used to calculate \( \cos(15^\circ) \)?
4. How does the value of \( \cos(105^\circ) \) compare to \( \cos(75^\circ) \)?
5. Why does breaking down angles into known values help in finding exact trigonometric values?

**Tip:** Remember, addition and subtraction formulas are especially helpful for angles that aren't common in the unit circle but can be decomposed into known angles.

Use an Addition or Subtraction Formula to find the exact value of the expression, cos(105°).

Learn how to find the exact value of cos(105°) using the cosine addition formula. This solution demonstrates step-by-step calculations, using the known values for cos(60°), cos(45°), sin(60°), and sin(45°). Suitable for high school trigonometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation