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 33

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3.sin(2θ + 10°) = cos(3θ - 20°)

Verified step by step guidance

Use the identity \( \sin(\alpha) = \cos(90^\circ - \alpha) \) to rewrite the equation: \( \sin(2\theta + 10^\circ) = \cos(3\theta - 20^\circ) \) becomes \( \sin(2\theta + 10^\circ) = \sin(90^\circ - (3\theta - 20^\circ)) \).

Simplify the right side: \( \sin(90^\circ - (3\theta - 20^\circ)) = \sin(110^\circ - 3\theta) \).

Set the arguments of the sine functions equal to each other: \( 2\theta + 10^\circ = 110^\circ - 3\theta \).

Solve for \( \theta \) by first adding \( 3\theta \) to both sides: \( 2\theta + 3\theta + 10^\circ = 110^\circ \).

Combine like terms and solve for \( \theta \): \( 5\theta + 10^\circ = 110^\circ \), then subtract \( 10^\circ \) from both sides and divide by 5.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant to this problem is the co-function identity, which states that sin(x) = cos(90° - x). This identity allows us to relate sine and cosine functions, facilitating the solving of equations that involve both.

Angle Addition Formulas

Angle addition formulas express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles. For example, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas are essential for simplifying expressions like sin(2θ + 10°) and cos(3θ - 20°) in the given equation, making it easier to find solutions.

Acute Angles

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