1. What is the Complementary Angle Identity?

The complementary angle identity states that the sine of an angle's complement equals the cosine of the angle itself. This fundamental trigonometric identity demonstrates the relationship between sine and cosine functions for complementary angles.

2. How Does the Calculator Work?

The calculator uses the complementary angle identity:

Where:

• \( \theta \) — The input angle (in degrees or radians)
• \( 90° - \theta \) — The complementary angle (when using degrees)
• \( \frac{\pi}{2} - \theta \) — The complementary angle (when using radians)

Explanation: This identity shows the fundamental relationship between sine and cosine functions for angles that add up to 90° (or π/2 radians).

3. Importance of Complementary Angles

Details: Understanding complementary angles is crucial in trigonometry as it simplifies calculations and reveals fundamental relationships between trigonometric functions. This identity is particularly useful in solving trigonometric equations and proving other identities.

4. Using the Calculator

Tips: Enter your angle value and select whether it's in degrees or radians. The calculator will compute both sin(90°-θ) and cos(θ) to demonstrate their equality.

5. Frequently Asked Questions (FAQ)

Q1: Why do sin(90°-θ) and cos(θ) equal each other?
A: This comes from the definitions of sine and cosine in right triangles, where the sine of one acute angle equals the cosine of the other acute angle.

Q2: Does this identity work for all angle measures?
A: Yes, the identity holds true for all real values of θ, not just acute angles.

Q3: What's the equivalent identity for cosine?
A: The complementary identity for cosine is: cos(90°-θ) = sin(θ).

Q4: How is this useful in real-world applications?
A: This identity is used in physics, engineering, and navigation to simplify calculations involving periodic phenomena.

Q5: What about negative angles?
A: The identity still holds for negative angles, as trigonometric functions are periodic.