Find one solution for each equation. Assume all angles involved are acute angles. cos(3θ + 11°) = sin( 7θ + 40°) 5 10

Verified step by step guidance

1. Recall the co-function identity in trigonometry: \(\cos A = \sin B\) implies that either \(A = B\) or \(A = 90^\circ - B\) (considering acute angles).
2. Set up the first equation by equating the angles directly: \(3\theta + 11^\circ = 7\theta + 40^\circ\).
3. Solve the equation from step 2 for \(\theta\) by isolating \(\theta\) on one side.
4. Set up the second equation using the complementary angle relationship: \(3\theta + 11^\circ = 90^\circ - (7\theta + 40^\circ)\).
5. Solve the equation from step 4 for \(\theta\) by simplifying and isolating \(\theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. For instance, the identity sin(x) = cos(90° - x) can be useful in transforming equations to find solutions involving sine and cosine functions.

Angle Addition Formulas

Angle addition formulas allow us to express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles. For example, cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and sin(A + B) = sin(A)cos(B) + cos(A)sin(B). These formulas are essential for simplifying expressions like cos(3θ + 11°) and sin(7θ + 40°) in the given equation.

Acute Angles in Trigonometry

Acute angles are angles that measure less than 90 degrees. In trigonometry, the values of sine and cosine for acute angles are always positive, which is important when solving equations involving these functions. Understanding the behavior of trigonometric functions in the context of acute angles helps in determining valid solutions for the given equation.