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 6.3

Textbook Question

Which one of the following equations has solution 3π/4
a. arctan 1 = x
b. arcsin √2/2 = x
c. arccos (―√2 /2) = x

Verified step by step guidance

Step 1: Understand the problem. We need to determine which inverse trigonometric function equation has a solution of \( \frac{3\pi}{4} \).

Step 2: Analyze option (a): \( \arctan(1) = x \). The \( \arctan \) function returns the angle whose tangent is 1. Recall that \( \tan(\frac{\pi}{4}) = 1 \), so \( \arctan(1) = \frac{\pi}{4} \). This does not match \( \frac{3\pi}{4} \).

Step 3: Analyze option (b): \( \arcsin(\frac{\sqrt{2}}{2}) = x \). The \( \arcsin \) function returns the angle whose sine is \( \frac{\sqrt{2}}{2} \). Recall that \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), so \( \arcsin(\frac{\sqrt{2}}{2}) = \frac{\pi}{4} \). This does not match \( \frac{3\pi}{4} \).

Step 4: Analyze option (c): \( \arccos(-\frac{\sqrt{2}}{2}) = x \). The \( \arccos \) function returns the angle whose cosine is \(-\frac{\sqrt{2}}{2} \). Recall that \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \), so \( \arccos(-\frac{\sqrt{2}}{2}) = \frac{3\pi}{4} \).

Step 5: Conclude that option (c) is the correct equation since it matches the given solution \( \frac{3\pi}{4} \).

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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 arctan, arcsin, and arccos, are used to find angles when given a ratio of sides in a right triangle. For example, arctan(1) gives the angle whose tangent is 1, which corresponds to π/4 radians. Understanding these functions is crucial for solving equations that involve finding angles from given trigonometric values.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and coordinates in a circular context. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine and the y-coordinate represents the sine of the angle. Knowing the coordinates of key angles, such as 3π/4, helps in determining the values of trigonometric functions and their inverses.

Trigonometric Values at Key Angles

Certain angles, like π/4, π/3, and π/6, have well-known sine, cosine, and tangent values. For instance, sin(3π/4) equals √2/2 and cos(3π/4) equals -√2/2. Recognizing these values allows for quick identification of the correct inverse function that yields a specific angle, which is essential for solving the given equations.