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

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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Problem 6.23

Textbook Question

Find the exact value of each real number y if it exists. Do not use a calculator.
y = arccos (―√3/2)

Verified step by step guidance

Understand that \( y = \arccos(-\sqrt{3}/2) \) means we are looking for an angle \( y \) whose cosine is \(-\sqrt{3}/2\).

Recall that the range of the \( \arccos \) function is \([0, \pi] \), which means \( y \) must be within this interval.

Identify the reference angle whose cosine is \( \sqrt{3}/2 \). This angle is \( \pi/6 \) or 30 degrees.

Since the cosine is negative, \( y \) must be in the second quadrant where cosine values are negative.

Determine the angle in the second quadrant by subtracting the reference angle from \( \pi \), i.e., \( y = \pi - \pi/6 \).

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos, are used to find the angle whose cosine is a given value. For example, if y = arccos(x), then cos(y) = x. Understanding these functions is crucial for solving problems that involve finding angles from known trigonometric ratios.

Range of the arccos Function

The arccos function has a specific range of values, which is [0, π] for real numbers. This means that when you calculate arccos for a value, the result will always be an angle between 0 and π radians. Recognizing this range helps in determining the possible values of y when solving equations involving arccos.

Trigonometric Values of Special Angles

Certain angles have known trigonometric values, which are often derived from the unit circle. For instance, cos(5π/6) = -√3/2. Familiarity with these special angles allows for quick identification of exact values when solving trigonometric equations, such as the one presented in the question.