Find each exact function value. ​See Example 3.​
tan π/4

Verified step by step guidance

1. Recognize that $\tan \theta =\frac{\sin \theta }{\cos \theta }$.
2. Identify that $\theta =\frac{\pi }{4}$ is a special angle in trigonometry.
3. Recall the values of sine and cosine for $\frac{\pi }{4}$: $\sin \frac{\pi }{4}=\frac{\sqrt{2}}{2}$ and $\cos \frac{\pi }{4}=\frac{\sqrt{2}}{2}$.
4. Substitute these values into the tangent formula: $\tan \frac{\pi }{4}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$.
5. Simplify the expression to find the exact value of $\tan \frac{\pi }{4}$.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. They are fundamental in trigonometry and are defined for angles measured in radians or degrees. For example, the tangent function is the ratio of the opposite side to the adjacent side in a right triangle.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a crucial tool in trigonometry for defining the values of trigonometric functions for all angles. Each point on the unit circle corresponds to an angle, allowing for the determination of sine, cosine, and tangent values based on the coordinates of that point.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of functions like sine, cosine, and tangent at specific angles, often expressed in terms of square roots or fractions. For instance, tan(π/4) equals 1, as it represents the ratio of the lengths of the opposite and adjacent sides of a right triangle where both sides are equal. Knowing these exact values is essential for solving trigonometric equations and problems.