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

1. Measuring Angles

Angles in Standard Position

Trigonometry

1. Measuring Angles

Angles in Standard Position

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1:44 minutes

Problem 36

Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.sec θ = 1.1606249

Verified step by step guidance

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \cos \theta = \frac{1}{\sec \theta} \).

Substitute the given value into the equation: \( \cos \theta = \frac{1}{1.1606249} \).

Calculate \( \cos \theta \) using the reciprocal of the given secant value.

Use the inverse cosine function to find \( \theta \): \( \theta = \cos^{-1}(\text{calculated value}) \).

Ensure that the calculated \( \theta \) is within the interval \([0^\circ, 90^\circ)\).

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Key Concepts

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

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Understanding this relationship is crucial for solving equations involving secant, as it allows us to convert secant values into cosine values, which can then be used to find the angle θ.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine (cos⁻¹), are used to find angles when given a trigonometric ratio. For example, if we have a cosine value, we can use the arccos function to determine the corresponding angle. This concept is essential for solving the equation sec(θ) = 1.1606249 by first converting it to cos(θ) and then applying the inverse function.

Angle Measurement in Degrees

In trigonometry, angles can be measured in degrees or radians. The problem specifies that the answer should be in decimal degrees, which means understanding how to convert between radians and degrees may be necessary. Additionally, knowing the range of angles (0° to 90°) helps in determining the correct quadrant for the solution.