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

Verified step by step guidance

1. Recall the definition of the tangent function in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
2. Identify the angle given: \(\theta = \frac{\pi}{4}\) radians, which corresponds to 45 degrees.
3. Use the known exact values for sine and cosine at \(\frac{\pi}{4}\): \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) 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 fraction to find the exact value of \(\tan \frac{\pi}{4}\).

Key Concepts

Understanding Radian Measure

Radian measure is a way to express angles based on the radius of a circle. One radian is the angle subtended by an arc equal in length to the radius. Knowing that π radians equal 180 degrees helps convert and interpret angles like π/4, which corresponds to 45 degrees.

Definition of the Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. On the unit circle, tan(θ) equals sin(θ) divided by cos(θ). Understanding this ratio is essential to find exact values of tangent for special angles like π/4.

Exact Values of Trigonometric Functions for Special Angles

Certain angles, such as π/6, π/4, and π/3, have well-known exact sine, cosine, and tangent values. For π/4 (45 degrees), both sine and cosine are √2/2, making tan(π/4) equal to 1. Memorizing these values aids in quickly solving trigonometric problems without a calculator.