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

1. Start by recalling the definitions of secant and cosecant: \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
2. Set the given equation \( \sec(3\beta + 10^\circ) = \csc(\beta + 8^\circ) \) in terms of sine and cosine: \( \frac{1}{\cos(3\beta + 10^\circ)} = \frac{1}{\sin(\beta + 8^\circ)} \).
3. Cross-multiply to eliminate the fractions: \( \sin(\beta + 8^\circ) = \cos(3\beta + 10^\circ) \).
4. Use the identity \( \sin(\theta) = \cos(90^\circ - \theta) \) to rewrite the equation: \( \cos(90^\circ - (\beta + 8^\circ)) = \cos(3\beta + 10^\circ) \).
5. 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 \( \beta \).

Key Concepts

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.