Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2.tan 25.4°

Verified step by step guidance

1. Recall the cofunction identity for tangent: \( \tan(\theta) = \cot(90^\circ - \theta) \).
2. Identify the given angle \( \theta = 25.4^\circ \).
3. Substitute \( \theta \) into the cofunction identity: \( \tan(25.4^\circ) = \cot(90^\circ - 25.4^\circ) \).
4. Calculate the angle: \( 90^\circ - 25.4^\circ = 64.6^\circ \).
5. Express \( \tan(25.4^\circ) \) in terms of its cofunction: \( \tan(25.4^\circ) = \cot(64.6^\circ) \).

Key Concepts

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles. For acute angles, the sine of an angle is equal to the cosine of its complement, and similarly for tangent and cotangent. For example, sin(θ) = cos(90° - θ) and tan(θ) = cot(90° - θ). Understanding these identities is crucial for rewriting functions in terms of their cofunctions.

Acute Angles

Acute angles are angles that measure less than 90 degrees. In trigonometry, the properties and values of trigonometric functions are often defined specifically for acute angles, as they yield positive values. Recognizing that the question specifies acute angles helps in applying the correct cofunction identities without concern for negative values or undefined functions.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, and their reciprocals, are fundamental in trigonometry. Each function relates the angles of a triangle to the ratios of its sides. For example, the tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. Understanding these functions is essential for manipulating and transforming them into their cofunction forms.