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

Verified step by step guidance

1. Recall the definition of a cofunction: for an angle \( \theta \), the cofunction identity states that \( \sin(90^\circ - \theta) = \cos \theta \) and similarly for other trigonometric functions, such as \( \csc(\theta) = \sec(90^\circ - \theta) \).
2. Identify the given function: \( \csc 25^\circ \) is the cosecant of 25 degrees, which is the reciprocal of sine, i.e., \( \csc \theta = \frac{1}{\sin \theta} \).
3. Use the cofunction identity for cosecant: \( \csc \theta = \sec(90^\circ - \theta) \). This means \( \csc 25^\circ = \sec(90^\circ - 25^\circ) \).
4. Calculate the complementary angle inside the cofunction: \( 90^\circ - 25^\circ = 65^\circ \).
5. Write the cofunction with the same value as the original expression: \( \csc 25^\circ = \sec 65^\circ \).

Key Concepts

Cosecant Function (csc)

The cosecant function is the reciprocal of the sine function, defined as csc θ = 1/sin θ. It is used to find the ratio of the hypotenuse to the opposite side in a right triangle. Understanding csc is essential to relate it to other trigonometric functions.

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, such as sin(90° - θ) = cos θ. These identities help find equivalent expressions by switching between pairs like sine and cosine or tangent and cotangent for angles summing to 90°.

Complementary Angles

Complementary angles are two angles whose measures add up to 90°. In trigonometry, many function values at an angle θ correspond to cofunctions at 90° - θ, enabling simplification or transformation of expressions using cofunction identities.