How to Find the Exact Values of Trigonometric Functions

Understanding and calculating the exact values of trigonometric functions is a fundamental skill in mathematics. Whether you're dealing with common angles such as 0^circ, 30^circ, 45^circ, 60^circ, and 90^circ or more complex angles, these methods will help you find their values accurately.

Common Angles

The first step in finding the exact values of trigonometric functions for common angles is to memorize the values of sine, cosine, and tangent. This is particularly useful when working with basic trigonometric problems.

Sine Values

sin0^circ 0 sin30^circ frac{1}{2} sin45^circ frac{sqrt{2}}{2} sin60^circ frac{sqrt{3}}{2} sin90^circ 1

Cosine Values

cos0^circ 1 cos30^circ frac{sqrt{3}}{2} cos45^circ frac{sqrt{2}}{2} cos60^circ frac{1}{2} cos90^circ 0

Tangent Values

tan0^circ 0 tan30^circ frac{1}{sqrt{3}} frac{sqrt{3}}{3} tan45^circ 1 tan60^circ sqrt{3} tan90^circ is undefined.

Unit Circle

The unit circle is a powerful tool for finding the values of trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane.

The coordinates of any point on the unit circle are given by (costheta, sintheta). For any angle theta, the sine and cosine values are the y-coordinate and x-coordinate, respectively. The tangent is found by the ratio tantheta frac{sintheta}{costheta}.

Reference Angles

When dealing with angles outside the range of 0^circ to 90^circ, the concept of reference angles is very useful. A reference angle is the acute angle formed with the x-axis. Depending on the quadrant, the signs of the trigonometric functions can be determined.

Quadrant I

Angles in Quadrant I are between 0^circ and 90^circ. All trigonometric functions are positive.

Quadrant II

Angles in Quadrant II are between 90^circ and 180^circ. Sine is positive, cosine and tangent are negative.

Quadrant III

Angles in Quadrant III are between 180^circ and 270^circ. Tangent is positive, sine and cosine are negative.

Quadrant IV

Angles in Quadrant IV are between 270^circ and 360^circ. Cosine is positive, sine and tangent are negative.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They can be used to find exact values of trigonometric functions in various scenarios.

Pythagorean Identity

The Pythagorean Identity is a fundamental identity in trigonometry that relates sine and cosine: sin^2theta cos^2theta 1.

Double Angle Formulas

Double angle formulas are identities that express trigonometric functions of double angles in terms of the functions of the single angle. For example, the double angle formula for sine is: sin2theta 2sinthetacostheta.

Example Problem

Let's find the exact value of sin135^circ.

First, find the reference angle: (theta 180^circ - 135^circ 45^circ).

The reference angle is 45^circ, and in the second quadrant, sine is positive. Therefore, sin135^circ sin45^circ frac{sqrt{2}}{2}.

Summary

To find the exact values of trigonometric functions, follow these steps:

Memorize the values for common angles Use the unit circle to find values for any angle Apply reference angles and quadrant signs Utilize trigonometric identities as needed

If you have a specific angle in mind, feel free to ask for its exact value!