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 6.7

Textbook Question

Solve each equation for x, where x is restricted to the given interval.
y = 5 cos x , for x in [0, π]

Verified step by step guidance

Step 1: Start with the equation y = 5 \cos(x).

Step 2: Divide both sides of the equation by 5 to isolate \cos(x), resulting in \cos(x) = \frac{y}{5}.

Step 3: Determine the values of x for which \cos(x) = \frac{y}{5} within the interval [0, \pi].

Step 4: Use the inverse cosine function to find x, i.e., x = \cos^{-1}\left(\frac{y}{5}\right).

Step 5: Verify that the solution x is within the interval [0, \pi] and adjust if necessary.

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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, denoted as cos(x), is a fundamental trigonometric function that relates the angle x to the ratio of the adjacent side to the hypotenuse in a right triangle. It oscillates between -1 and 1, with a period of 2π. Understanding its behavior is crucial for solving equations involving cosine, especially within specified intervals.

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. The solution must also respect any given restrictions on the variable, such as the interval [0, π].

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, [0, π] indicates that x can take any value from 0 to π, inclusive. Understanding interval notation is essential for determining valid solutions to the equation, as it restricts the possible values of x to a specific range.