https://file.aiquickdraw.com/mathbot/file/1729749813181sdaywxn9.jpg

Let's analyze the given trigonometric expressions to determine how many have known exact values due to being related to special angles:

1. **cos 30°**: This is related to the special angle 30°, and its exact value is \( \frac{\sqrt{3}}{2} \).
2. **sin 45°**: This is a special angle (45°), and its exact value is \( \frac{\sqrt{2}}{2} \).
3. **sin 405°**: This angle can be reduced by subtracting 360° (since sine functions are periodic with period 360°). \( 405^\circ - 360^\circ = 45^\circ \), so sin 405° = sin 45°, which has the known exact value \( \frac{\sqrt{2}}{2} \).
4. **cos 315°**: Subtracting 360° gives \( 315^\circ - 360^\circ = -45^\circ \). Using the cosine of negative angles property, \( \cos(-45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2} \), so it has a known exact value.
5. **tan -45°**: This is also a special angle, and its exact value is \( -1 \).

Thus, **all five** of these expressions are related to special angles and have known exact values.

The correct answer should be **5**, not 3.

Would you like any further clarification or details? Here are some follow-up questions:

1. How do you simplify trigonometric expressions for angles greater than 360° or less than 0°?
2. Why are 30°, 45°, and 60° called special angles in trigonometry?
3. How do you find the exact value of trigonometric functions for negative angles?
4. What is the periodicity of sine, cosine, and tangent functions?
5. How can reference angles help in finding trigonometric values for angles beyond 360°?

**Tip**: Always try to reduce angles to within 0° to 360° using their periodic properties for easier evaluation of trigonometric values.

How many of the following items have known exact values because they are related to special angles? (cos 30°, sin 45°, sin 405°, cos 315°, tan −45°)

Explore how to determine the exact values of trigonometric functions for special angles, including cos 30°, sin 45°, sin 405°, cos 315°, and tan -45°. Learn about periodicity and exact values for these angles, suitable for high school 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