Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Verified step by step guidance

1. Step 1: Identify the given information. If Example 1 provides a specific angle or triangle, note the values or relationships given.
2. Step 2: Use the Pythagorean identity to find missing values. Recall that \( \sin^2 A + \cos^2 A = 1 \). If one of \( \sin A \) or \( \cos A \) is given, solve for the other.
3. Step 3: Calculate \( \tan A \) using the identity \( \tan A = \frac{\sin A}{\cos A} \).
4. Step 4: If the problem involves a right triangle, use SOH-CAH-TOA to find \( \sin A \), \( \cos A \), and \( \tan A \) based on the sides of the triangle.
5. Step 5: Verify your results by checking if they satisfy the trigonometric identities and the context of the problem.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are the relationships between the angles and sides of a right triangle. The primary ratios are sine (sin), cosine (cos), and tangent (tan), defined as sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, and tan A = opposite/adjacent. Understanding these ratios is essential for finding the exact values of sin A, cos A, and tan A for a given angle.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric representation of the trigonometric functions, where the coordinates of any point on the circle correspond to the values of cos and sin for the angle formed with the positive x-axis. This concept is crucial for determining exact values of trigonometric functions for common angles.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle in standard position and the x-axis. They help in finding the sine, cosine, and tangent values for angles greater than 90 degrees or less than 0 degrees by relating them back to their corresponding acute angles. This concept is important for accurately determining the values of sin A, cos A, and tan A in various quadrants.