In Exercises 31–38, find a cofunction with the same value as the given expression.csc 25°

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1. Understand that cofunctions are pairs of trigonometric functions that are equal when their angles are complementary. The cofunction identities are: \( \sin(90^\circ - \theta) = \cos(\theta) \), \( \cos(90^\circ - \theta) = \sin(\theta) \), \( \tan(90^\circ - \theta) = \cot(\theta) \), \( \cot(90^\circ - \theta) = \tan(\theta) \), \( \sec(90^\circ - \theta) = \csc(\theta) \), and \( \csc(90^\circ - \theta) = \sec(\theta) \).
2. Identify the given expression: \( \csc 25^\circ \).
3. Use the cofunction identity for cosecant: \( \csc(\theta) = \sec(90^\circ - \theta) \).
4. Substitute \( \theta = 25^\circ \) into the cofunction identity: \( \csc(25^\circ) = \sec(90^\circ - 25^\circ) \).
5. Simplify the expression: \( \sec(65^\circ) \). Therefore, \( \sec 65^\circ \) is the cofunction with the same value as \( \csc 25^\circ \).

Key Concepts

Cofunction Identities

Cofunction identities in trigonometry relate the values of trigonometric functions of complementary angles. Specifically, for any angle θ, the sine of θ is equal to the cosine of its complement (90° - θ). This means that functions like sine, cosine, tangent, and cotangent have corresponding cofunctions that can be used to find equivalent values.

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. For an angle θ, csc(θ) = 1/sin(θ). Understanding this relationship is crucial for finding cofunctions, as it allows us to express csc(25°) in terms of another trigonometric function that can be evaluated using complementary angles.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In the context of trigonometric functions, knowing the complementary angle allows us to use cofunction identities effectively. For example, since csc(25°) relates to sin(65°) (where 65° is the complement of 25°), this relationship is key to finding the cofunction with the same value.