Find each exact function value.
tan π/3

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: \(\theta = \frac{\pi}{3}\) radians, which corresponds to 60 degrees.
3. Use the known exact values for sine and cosine at \(\frac{\pi}{3}\): \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\) and \(\cos \frac{\pi}{3} = \frac{1}{2}\).
4. Substitute these values into the tangent formula: \(\tan \frac{\pi}{3} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\).
5. Simplify the fraction by dividing the numerators and denominators accordingly to find the exact value of \(\tan \frac{\pi}{3}\).

Key Concepts

Understanding the Unit Circle

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. It helps define trigonometric functions for all angles by relating angles to coordinates (x, y), where x = cos(θ) and y = sin(θ). Knowing the unit circle values for common angles like π/3 is essential for finding exact trigonometric values.

Definition of the Tangent Function

Tangent of an angle θ in a right triangle is the ratio of the opposite side to the adjacent side. Using the unit circle, tan(θ) can be expressed as sin(θ)/cos(θ). This ratio allows calculation of tangent values from known sine and cosine values.

Special Angles and Their Exact Values

Special angles such as π/6, π/4, and π/3 have well-known exact sine, cosine, and tangent values derived from equilateral and isosceles triangles. For π/3, sin(π/3) = √3/2 and cos(π/3) = 1/2, which are used to find tan(π/3) exactly.