Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7.
cos θ = 1

Verified step by step guidance

1. Identify the given trigonometric function and its value: here, \( \cos \theta = 1 \).
2. Recall the definition of cosine in terms of the unit circle: \( \cos \theta = \frac{x}{r} \), where \( x \) is the horizontal coordinate and \( r = 1 \) on the unit circle.
3. Determine the angle(s) \( \theta \) where \( \cos \theta = 1 \). On the unit circle, this occurs at \( \theta = 0 \) (or multiples of \( 2\pi \)).
4. Use the Pythagorean identity to find \( \sin \theta \): \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = 1 \) to find \( \sin \theta \).
5. Calculate the remaining trigonometric functions using their definitions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \). Rationalize denominators if needed.

Key Concepts

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of sides in a right triangle or coordinates on the unit circle. Given one function value, the others can be found using their interrelationships and identities.

Unit Circle and Angle Interpretation

The unit circle represents angles as points (x, y) where x = cos θ and y = sin θ. Knowing cos θ = 1 corresponds to the point (1, 0), which helps determine all other function values for that angle.

Rationalizing Denominators

Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This is often required for final answers to be in simplified, standard form.