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

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Recognize that the equation involves trigonometric functions of angles: \( \cos(3\theta + 11^\circ) = \sin(\frac{7\theta + 40^\circ}{10}) \).

Use the identity \( \cos A = \sin B \) which implies \( A = 90^\circ - B \) to set up the equation: \( 3\theta + 11^\circ = 90^\circ - \frac{7\theta + 40^\circ}{10} \).

Rearrange the equation to isolate terms involving \( \theta \): \( 3\theta + 11^\circ + \frac{7\theta + 40^\circ}{10} = 90^\circ \).

Multiply through by 10 to eliminate the fraction: \( 10(3\theta + 11^\circ) + (7\theta + 40^\circ) = 900^\circ \).

Combine like terms and solve for \( \theta \) to find one solution.

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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. 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.

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