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 40

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3.csc(β + 40°) = sec(β - 20°)

Verified step by step guidance

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

Rewrite the given equation \( \csc(\beta + 40^\circ) = \sec(\beta - 20^\circ) \) using these definitions: \( \frac{1}{\sin(\beta + 40^\circ)} = \frac{1}{\cos(\beta - 20^\circ)} \).

Cross-multiply to eliminate the fractions: \( \cos(\beta - 20^\circ) = \sin(\beta + 40^\circ) \).

Use the co-function identity \( \sin(\theta) = \cos(90^\circ - \theta) \) to rewrite \( \sin(\beta + 40^\circ) \) as \( \cos(50^\circ - \beta) \).

Set the expressions equal: \( \cos(\beta - 20^\circ) = \cos(50^\circ - \beta) \), and solve for \( \beta \) using the property that if \( \cos(A) = \cos(B) \), then \( A = B + 360^\circ k \) or \( A = -B + 360^\circ k \) for integer \( k \).

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Key Concepts

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

Cosecant and Secant Functions

Cosecant (csc) and secant (sec) are reciprocal trigonometric functions. Cosecant is the reciprocal of sine, defined as csc(θ) = 1/sin(θ), while secant is the reciprocal of cosine, defined as sec(θ) = 1/cos(θ). Understanding these functions is crucial for solving equations involving them, as it allows for the transformation of the equation into a more manageable form.

Angle Addition Formulas

The angle addition formulas are essential for simplifying expressions involving sums of angles. For example, sin(α + β) = sin(α)cos(β) + cos(α)sin(β) and cos(α + β) = cos(α)cos(β) - sin(α)sin(β). In the given equation, recognizing that csc(β + 40°) and sec(β - 20°) can be expressed using these formulas will facilitate solving for the angle β.

Acute Angles in Trigonometry

Acute angles are angles that measure less than 90 degrees. In trigonometry, the values of sine, cosine, and tangent for acute angles are always positive. This property is important when solving equations, as it restricts the possible solutions to those that fall within the first quadrant, ensuring that the angles involved in the equation remain acute.