Find a cofunction with the same value as the given expression.
sin 19°

Verified step by step guidance

1. Recall the cofunction identity for sine and cosine: \(\sin(\theta) = \cos(90^\circ - \theta)\).
2. Identify the angle in the given expression, which is \(19^\circ\) in \(\sin 19^\circ\).
3. Apply the cofunction identity by substituting \(\theta = 19^\circ\) into the formula: \(\sin 19^\circ = \cos(90^\circ - 19^\circ)\).
4. Simplify the expression inside the cosine function: \(90^\circ - 19^\circ = 71^\circ\).
5. Conclude that the cofunction with the same value as \(\sin 19^\circ\) is \(\cos 71^\circ\).

Key Concepts

Trigonometric Cofunction Identity

Cofunction identities relate pairs of trigonometric functions whose angles add up to 90°. For example, sin(θ) equals cos(90° - θ). This means sin 19° can be expressed as cos(71°), since 19° + 71° = 90°.

Sine Function

The sine function gives the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. It is periodic and ranges between -1 and 1, and is fundamental in relating angles to side lengths.

Complementary Angles

Two angles are complementary if their sum is 90°. In trigonometry, complementary angles are important because the sine of one angle equals the cosine of its complement, enabling the use of cofunction identities.