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

검증된 단계별 안내

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.

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 \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\).

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.

Substitute the known side lengths into the formulas for \(\sin A\), \(\cos A\), and \(\tan A\) to write expressions for each function.

Simplify the expressions if possible, and express the values exactly (e.g., in terms of square roots or fractions) rather than decimal approximations.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.