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

Verified step by step guidance

1. Recall the identity that relates sine and cosecant: \(\csc x = \frac{1}{\sin x}\). Using this, rewrite the equation \(\sin(4\theta + 2^\circ) \cdot \csc(3\theta + 5^\circ) = 1\) as \(\sin(4\theta + 2^\circ) \cdot \frac{1}{\sin(3\theta + 5^\circ)} = 1\).
2. Simplify the equation to get \(\frac{\sin(4\theta + 2^\circ)}{\sin(3\theta + 5^\circ)} = 1\).
3. From the simplified equation, deduce that \(\sin(4\theta + 2^\circ) = \sin(3\theta + 5^\circ)\).
4. Use the general solution for when two sine values are equal: \(\sin A = \sin B\) implies \(A = B + 360^\circ k\) or \(A = 180^\circ - B + 360^\circ k\), where \(k\) is any integer.
5. Set up the two equations based on the above and solve for \(\theta\) in each case: 1) \(4\theta + 2^\circ = 3\theta + 5^\circ + 360^\circ k\) 2) \(4\theta + 2^\circ = 180^\circ - (3\theta + 5^\circ) + 360^\circ k\). Then isolate \(\theta\) in each equation to find the general solutions.

Key Concepts

Relationship Between Sine and Cosecant

Cosecant is the reciprocal of sine, meaning csc(x) = 1/sin(x). Understanding this relationship allows you to rewrite the equation sin(A) * csc(B) = 1 as sin(A) / sin(B) = 1, which simplifies the problem and helps in finding solutions for the angles involved.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. This often requires using identities, inverse functions, and considering the periodic nature of sine and cosecant to find general solutions.

Angle Measures and Periodicity

Trigonometric functions are periodic, meaning their values repeat at regular intervals (360° for sine and cosecant). When solving equations like sin(4θ + 2°) = sin(3θ + 5°), it is essential to consider all possible angles by adding integer multiples of the period to find the complete set of solutions.