Give the exact value of each expression. See Example 5. tan 30°

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 values of \(\sin 30^\circ\) and \(\cos 30^\circ\) using known special angles: \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
3. Substitute these values into the tangent formula: \(\tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\).
4. Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\tan 30^\circ = \frac{1}{2} \times \frac{2}{\sqrt{3}}\).
5. Further simplify the expression to get the exact value of \(\tan 30^\circ\) in simplest radical form.

Key Concepts

Definition of the Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. It can also be expressed as tan(θ) = sin(θ)/cos(θ), linking it to the sine and cosine functions.

Special Angles and Their Exact Values

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values derived from special triangles. For 30°, these values come from the 30°-60°-90° triangle, enabling precise calculation without a calculator.

Using the 30°-60°-90° Triangle

The 30°-60°-90° triangle has side ratios of 1:√3:2. Knowing these ratios allows direct computation of trigonometric functions for 30°, such as tan 30° = opposite/adjacent = 1/√3, which can be rationalized to √3/3.