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

3. Unit Circle

Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Trigonometric Functions on the Unit Circle

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Problem 39

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3.sec(3β + 10°) = csc(β + 8°)

Verified step by step guidance

Start by recalling the definitions of secant and cosecant: \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Set the given equation \( \sec(3\beta + 10^\circ) = \csc(\beta + 8^\circ) \) in terms of sine and cosine: \( \frac{1}{\cos(3\beta + 10^\circ)} = \frac{1}{\sin(\beta + 8^\circ)} \).

Cross-multiply to eliminate the fractions: \( \sin(\beta + 8^\circ) = \cos(3\beta + 10^\circ) \).

Use the identity \( \sin(\theta) = \cos(90^\circ - \theta) \) to rewrite the equation: \( \cos(90^\circ - (\beta + 8^\circ)) = \cos(3\beta + 10^\circ) \).

Set the angles equal to each other, considering the periodicity of cosine: \( 90^\circ - (\beta + 8^\circ) = 3\beta + 10^\circ + 360^\circ k \), where \( k \) is an integer, and solve for \( \beta \).

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Key Concepts

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

Secant and Cosecant Functions

Secant (sec) and cosecant (csc) are trigonometric functions defined as the reciprocals of cosine and sine, respectively. Specifically, sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ). Understanding these functions is crucial for solving equations involving them, as it allows for the manipulation and transformation of the equation into a more solvable form.

Angle Addition

The angle addition formulas are essential in trigonometry for simplifying expressions involving sums of angles. For example, sec(3β + 10°) can be expressed using the angle addition formula, which helps in breaking down the equation into manageable parts. This concept is particularly useful when dealing with equations that involve multiple angles.

Acute Angles

Acute angles are angles that measure less than 90 degrees. In the context of this problem, assuming all angles are acute simplifies the analysis, as the values of sine and cosine are positive. This assumption is important when solving trigonometric equations, as it influences the range of possible solutions and the behavior of the trigonometric functions involved.