Find a cofunction with the same value as the given expression.
csc 35°

Verified step by step guidance

1. Recall the cofunction identity for cosecant: \( \csc \theta = \sec (90^\circ - \theta) \). This means that the cosecant of an angle is equal to the secant of its complement.
2. Identify the given angle \( \theta = 35^\circ \). To find the cofunction, calculate the complement of this angle: \( 90^\circ - 35^\circ = 55^\circ \).
3. Apply the cofunction identity: \( \csc 35^\circ = \sec 55^\circ \). This shows that \( \sec 55^\circ \) is the cofunction with the same value as \( \csc 35^\circ \).
4. Understand that this works because sine and cosine are cofunctions, and since \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \), the reciprocal relationship holds for their cofunctions as well.
5. Therefore, the expression \( \csc 35^\circ \) can be rewritten as \( \sec 55^\circ \), which is the cofunction with the same value.

Key Concepts

Cosecant Function (csc)

The cosecant function is the reciprocal of the sine function, defined as csc θ = 1/sin θ. It represents 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 allow expressing one function in terms of another with an angle complement, which is key to finding a cofunction with the same value.

Complementary Angles

Complementary angles sum to 90°, and many trigonometric identities use this property to connect functions. Recognizing that 35° and 55° are complementary helps apply cofunction identities to rewrite csc 35° in terms of another function.