Concept Check Find a solution for each equation. sin(4θ + 2°) csc(3θ + 5°) = 1

Verified step by step guidance

1. Recognize that \( \csc(x) = \frac{1}{\sin(x)} \), so rewrite the equation as \( \sin(4\theta + 2^\circ) \cdot \frac{1}{\sin(3\theta + 5^\circ)} = 1 \).
2. Simplify the equation to \( \frac{\sin(4\theta + 2^\circ)}{\sin(3\theta + 5^\circ)} = 1 \).
3. To satisfy the equation \( \frac{\sin(4\theta + 2^\circ)}{\sin(3\theta + 5^\circ)} = 1 \), set \( \sin(4\theta + 2^\circ) = \sin(3\theta + 5^\circ) \).
4. Use the identity that if \( \sin(A) = \sin(B) \), then \( A = B + 360^\circ k \) or \( A = 180^\circ - B + 360^\circ k \), where \( k \) is an integer.
5. Solve the resulting equations for \( \theta \) to find the general solutions.

Key Concepts

Sine and Cosecant Functions

The sine function, sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. The cosecant function, csc(θ), is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). Understanding these functions is crucial for solving equations involving trigonometric identities, as they often appear in various forms.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. Recognizing and applying these identities can simplify complex trigonometric equations, making it easier to find solutions.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation. This often requires isolating the trigonometric function and using inverse functions or identities to determine the angle solutions. Understanding the periodic nature of trigonometric functions is also essential, as solutions may repeat over specific intervals.