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

Previous problemNext problem

1:51 minutes

Problem 9

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.
cos 30°

Verified step by step guidance

Identify the sides of the triangle relative to the 30° angle: the hypotenuse is PR = 2, the adjacent side is QR = 1, and the opposite side is PQ = \( \sqrt{3} \).

Recall the definition of cosine for an angle in a right triangle: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Substitute the known values into the cosine formula: \( \cos(30°) = \frac{QR}{PR} = \frac{1}{2} \).

Since the value is already simplified, there is no need to rationalize the denominator.

Conclude that the expression for \( \cos(30°) \) is \( \frac{1}{2} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of a right triangle. The primary ratios include sine, cosine, and tangent, defined as the ratios of the lengths of the sides opposite, adjacent, and hypotenuse to a given angle. For example, in triangle PQR, cos(30°) is calculated as the length of the adjacent side (QR) over the hypotenuse (PR).

Special Right Triangles

Special right triangles, specifically the 30-60-90 triangle, have known side ratios that simplify calculations. In a 30-60-90 triangle, the sides opposite the 30°, 60°, and 90° angles are in the ratio 1:√3:2. This means that if the shortest side is 1, the longer leg is √3, and the hypotenuse is 2, making it easier to find trigonometric values without complex calculations.

Rationalizing the Denominator

Rationalizing the denominator is a 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. For example, if the expression involves a square root in the denominator, multiplying by that square root can help simplify the expression.