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 6.67

Textbook Question

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)
y = sec⁻¹ (―1.2871684)

Verified step by step guidance

Ensure your calculator is in radian mode, as the problem specifies the use of radians.

Understand that \( \sec^{-1}(x) \) is the inverse secant function, which gives the angle whose secant is \( x \).

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so \( \sec^{-1}(x) \) is equivalent to finding \( \theta \) such that \( \cos(\theta) = \frac{1}{x} \).

Calculate \( \cos(\theta) = \frac{1}{-1.2871684} \) to find the cosine of the angle.

Use the inverse cosine function on your calculator to find \( \theta = \cos^{-1}\left(\frac{1}{-1.2871684}\right) \).

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Key Concepts

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

Inverse Secant Function

The inverse secant function, denoted as sec⁻¹ or arcsec, is the inverse of the secant function. It is defined for values outside the interval (-1, 1), as sec(x) = 1/cos(x). The output of the inverse secant function is an angle whose secant is the given value, and it typically returns values in the range [0, π/2) ∪ (π/2, π].

Radian Mode

Radian mode is a setting on calculators that allows angles to be measured in radians instead of degrees. Radians are a unit of angular measure where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. This mode is essential for trigonometric calculations involving functions like sine, cosine, and secant, especially when working with calculus or higher-level mathematics.

Calculator Functions

Understanding how to use a scientific calculator is crucial for approximating values of trigonometric functions. Most calculators have specific buttons for inverse functions, including sec⁻¹. Familiarity with the calculator's interface and functions ensures accurate input and interpretation of results, especially when switching between radian and degree modes.