2sin 67.5° cos 67.5° write the following expressions as the sine, cosine, or tangent of a double angle. then find the exact value of the expression. simplify answer, answer in degree.

To solve \( 2 \sin 67.5^\circ \cos 67.5^\circ \), we use the double angle identity for sine:

\[
\sin(2x) = 2 \sin(x) \cos(x)
\]

Here, \( x = 67.5^\circ \). Substituting into the identity:

\[
2 \sin 67.5^\circ \cos 67.5^\circ = \sin(2 \times 67.5^\circ) = \sin(135^\circ)
\]

### Step 1: Find \( \sin(135^\circ) \) exactly
Using the unit circle, \( 135^\circ \) is in the second quadrant where sine is positive. The reference angle is \( 180^\circ - 135^\circ = 45^\circ \). Thus:

\[
\sin(135^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}
\]

### Final Answer:
\[
2 \sin 67.5^\circ \cos 67.5^\circ = \frac{\sqrt{2}}{2}
\]

Would you like further clarification or details?

Here are 5 related questions to deepen your understanding:

1. How does the double-angle identity for sine relate to the double-angle identities for cosine?
2. Can you derive the exact value of \( \sin(67.5^\circ) \) using half-angle identities?
3. How would the expression change if it were \( 2 \sin 67.5^\circ \cos 67.5^\circ - \cos^2(67.5^\circ) \)?
4. Why is it important to know the quadrant when finding the exact value of trigonometric functions?
5. How is the double-angle formula useful in solving real-world problems involving oscillations or waves?

**Tip:** Memorize the special angles and their trigonometric values; they frequently simplify calculations involving identities!

Learn how to simplify the expression 2sin(67.5°)cos(67.5°) using the double angle identity for sine. This step-by-step solution involves applying trigonometric identities and finding the exact value of sin(135°) based on special angles. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation