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 6.21

Textbook Question

Solve each equation for x, where x is restricted to the given interval.
y = cos (x + 3) , for x in [―3, π―3]

Verified step by step guidance

Step 1: Start by understanding the equation y = \cos(x + 3). We need to solve for x within the interval [-3, \pi - 3].

Step 2: Recognize that the cosine function is periodic with a period of 2\pi. This means that \cos(\theta) = \cos(\theta + 2k\pi) for any integer k.

Step 3: To solve for x, set \cos(x + 3) = y. This implies x + 3 = \cos^{-1}(y) + 2k\pi or x + 3 = -\cos^{-1}(y) + 2k\pi, where k is an integer.

Step 4: Solve for x by isolating it: x = \cos^{-1}(y) - 3 + 2k\pi or x = -\cos^{-1}(y) - 3 + 2k\pi.

Step 5: Determine the values of k such that x is within the interval [-3, \pi - 3]. Evaluate both expressions for x and check which values satisfy the interval condition.

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Key Concepts

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

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π, meaning it repeats its values every 2π units. Understanding the properties of the cosine function, including its range of values from -1 to 1, is essential for solving equations involving cosine.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions. In this case, we need to isolate x and consider the specified interval to find valid solutions.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [―3, π―3] indicates that x can take any value from -3 to π - 3, inclusive. Understanding how to interpret and apply interval notation is crucial for determining which solutions to the trigonometric equation are valid within the specified range.