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 6.15

Textbook Question

Solve each equation for x, where x is restricted to the given interval.
y = sin x ―2 , for x in [―π/2. π/2]

Verified step by step guidance

Step 1: Start with the given equation: \( y = \sin x - 2 \).

Step 2: Rearrange the equation to solve for \( \sin x \): \( \sin x = y + 2 \).

Step 3: Determine the range of \( \sin x \) which is \([-1, 1]\). Therefore, \( y + 2 \) must also be within this range.

Step 4: Solve the inequality \(-1 \leq y + 2 \leq 1\) to find the possible values of \( y \).

Step 5: Use the inverse sine function to solve for \( x \) within the interval \([-\pi/2, \pi/2]\) for the valid values of \( y \).

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Key Concepts

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

Sine Function

The sine function, denoted as sin(x), is a fundamental trigonometric function that relates the angle x (measured in radians) to the ratio of the length of the opposite side to the hypotenuse in a right triangle. It oscillates between -1 and 1, making it crucial for understanding periodic behavior in trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable (in this case, x) that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions to find all possible solutions within a specified interval.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [−π/2, π/2] indicates that x can take any value from −π/2 to π/2, inclusive. Understanding this notation is essential for determining the valid solutions to the equation within the specified bounds.