In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

Verified step by step guidance

1. Identify the coordinates of the point P(x, y) on the unit circle.
2. Recall that on the unit circle, the x-coordinate represents \( \cos(t) \) and the y-coordinate represents \( \sin(t) \).
3. Use the coordinates to find \( \cos(t) \) and \( \sin(t) \).
4. Calculate \( \tan(t) \) using the formula \( \tan(t) = \frac{\sin(t)}{\cos(t)} \), provided \( \cos(t) \neq 0 \).
5. Determine the values of the reciprocal trigonometric functions: \( \csc(t) = \frac{1}{\sin(t)} \), \( \sec(t) = \frac{1}{\cos(t)} \), and \( \cot(t) = \frac{1}{\tan(t)} \), ensuring the denominators are not zero.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine and cosine functions. Any point on the unit circle can be expressed as (cos(t), sin(t)), where t is the angle formed with the positive x-axis, allowing for easy calculation of trigonometric values.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. On the unit circle, the sine of an angle corresponds to the y-coordinate, while the cosine corresponds to the x-coordinate of a point on the circle. Understanding these functions is essential for solving problems involving angles and distances in trigonometry.

Angle Measurement

Angles in trigonometry can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. One complete revolution around the unit circle corresponds to 2π radians or 360 degrees. Knowing how to convert between these two units is crucial for accurately determining the values of trigonometric functions at specific angles.