Write each function value in terms of the cofunction of a complementary angle.
sin 142° 14'

Verified step by step guidance

1. Recall the cofunction identity for sine and cosine: \(\sin(\theta) = \cos(90^\circ - \theta)\), where the angles are complementary (sum to \(90^\circ\)).
2. Identify the given angle: \(142^\circ 14'\) is greater than \(90^\circ\), so it is not directly complementary to an angle between \(0^\circ\) and \(90^\circ\).
3. Use the periodicity and symmetry of sine to rewrite \(\sin 142^\circ 14'\) in terms of an angle between \(0^\circ\) and \(90^\circ\). For example, use the identity \(\sin(180^\circ - \alpha) = \sin \alpha\) to find an acute angle complementary to another angle.
4. Calculate the reference angle: \(180^\circ - 142^\circ 14' = 37^\circ 46'\). So, \(\sin 142^\circ 14' = \sin 37^\circ 46'\).
5. Now express \(\sin 37^\circ 46'\) as the cosine of its complementary angle: \(\sin 37^\circ 46' = \cos(90^\circ - 37^\circ 46') = \cos 52^\circ 14'\).

Key Concepts

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, where the sum of the angles is 90°. For example, sin(θ) = cos(90° - θ). This allows expressing one function in terms of the cofunction of its complementary angle.

Complementary Angles

Complementary angles are two angles whose measures add up to 90°. Understanding this is essential because cofunction identities depend on the relationship between an angle and its complement.

Angle Conversion and Notation

Angles can be expressed in degrees and minutes, where 1 degree = 60 minutes. Properly interpreting and converting these units is important for accurate calculation and application of trigonometric identities.