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

1. Measuring Angles

Radians

Trigonometry

1. Measuring Angles

Radians

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2:22 minutes

Problem 1

Textbook Question

Convert 135° to an exact radian measure.

Verified step by step guidance

Start by recalling the conversion factor between degrees and radians: \( \pi \) radians is equivalent to 180°.

Set up the conversion by multiplying 135° by the conversion factor \( \frac{\pi}{180} \).

Express the multiplication: \( 135° \times \frac{\pi}{180} \).

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 45.

The simplified expression will give you the exact radian measure of 135° in terms of \( \pi \).

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Key Concepts

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

Degree to Radian Conversion

To convert degrees to radians, use the conversion factor π radians = 180 degrees. This means that to convert an angle in degrees to radians, you multiply the degree measure by π/180. For example, to convert 135°, you would calculate 135 × (π/180).

Understanding Radians

Radians are a unit of angular measure used in mathematics, particularly in trigonometry. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. This makes radians a natural way to measure angles in relation to the properties of circles.

Exact Values in Trigonometry

Exact values in trigonometry refer to the precise values of trigonometric functions at specific angles, often expressed in terms of square roots and fractions. For example, the sine and cosine of 135° can be expressed as exact values involving √2, which is important for solving problems without a calculator.