Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

Trigonometry

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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1:37 minutes

Problem 10

Textbook Question

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
tan 30°

Verified step by step guidance

Identify the sides of the triangle relative to the 30° angle: opposite side is QR, adjacent side is PQ, and hypotenuse is PR.

Recall the definition of tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

For \( \tan(30^\circ) \), use the sides: opposite = 1, adjacent = \( \sqrt{3} \).

Set up the expression: \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).

Rationalize the denominator: multiply numerator and denominator by \( \sqrt{3} \) to get \( \frac{\sqrt{3}}{3} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric ratio defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. For example, in triangle PQR, tan(30°) can be calculated using the lengths of the sides opposite and adjacent to the 30° angle.

Special Right Triangles

Special right triangles, specifically the 30-60-90 triangle, have known side ratios: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is √3 times the shorter leg. In triangle PQR, the sides are labeled 1 (opposite 30°), 2 (hypotenuse), and √3 (opposite 60°), which helps in easily determining trigonometric values.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical process used to eliminate square roots or irrational numbers from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will result in a rational number in the denominator, ensuring the expression is in a standard form.