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 angle \( \theta \) for which \( \cos \theta = 1 \). This occurs when \( \theta = 0 \) degrees or \( \theta = 360 \) degrees (or \( \theta = 0 \) radians or \( \theta = 2\pi \) radians).
2. Use the angle \( \theta = 0 \) to find the sine function: \( \sin \theta = 0 \).
3. Determine the tangent function using \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0}{1} = 0 \).
4. Find the cosecant function, which is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \). Since \( \sin \theta = 0 \), \( \csc \theta \) is undefined.
5. Calculate the secant function, which is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{1} = 1 \), and the cotangent function, which is the reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \tan \theta = 0 \), \( \cot \theta \) is undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The six primary functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each function can be defined using a right triangle or the unit circle, providing a way to calculate the ratios of the sides based on the angle θ.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions for all angles, not just those in right triangles. The coordinates of points on the unit circle correspond to the values of cosine and sine for the angle, facilitating the calculation of all six trigonometric functions.

Rationalizing Denominators

Rationalizing the denominator is a mathematical process used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will result in a rational number in the denominator. In trigonometry, this is particularly important when dealing with function values that involve square roots, ensuring that the final answers are presented in a standard form.