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

Verified step by step guidance

1. Identify the given information about angle A, such as the sides of the right triangle or the coordinates on the unit circle, since sin A, cos A, and tan A depend on these values.
2. Recall the definitions of the trigonometric functions in a right triangle: \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\).
3. If the sides of the triangle are not given, use the Pythagorean theorem \(a^2 + b^2 = c^2\) to find the missing side lengths, where \(c\) is the hypotenuse.
4. Substitute the known side lengths into the formulas for \(\sin A\), \(\cos A\), and \(\tan A\) to write expressions for each function.
5. Simplify the expressions if possible, and express the values exactly (e.g., in terms of square roots or fractions) rather than decimal approximations.

Key Concepts

Definition of Sine, Cosine, and Tangent

Sine, cosine, and tangent are fundamental trigonometric functions relating the angles of a right triangle to the ratios of its sides. Specifically, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, and tan A = opposite/adjacent. Understanding these definitions is essential for finding exact values.

Using Reference Triangles and Special Angles

Reference triangles, especially those with angles of 30°, 45°, and 60°, provide exact trigonometric values. Recognizing these special angles and their side ratios allows for precise calculation of sin A, cos A, and tan A without a calculator.

Trigonometric Identities and Relationships

Key identities like tan A = sin A / cos A and the Pythagorean identity sin² A + cos² A = 1 help relate the functions and verify results. These relationships are useful for expressing one function in terms of others and simplifying expressions.