Acute Angles Write the following function in terms of its cofunction. Assume that all angles in which an unknown appears are acute angles. sin (θ + 25°) ___ sin (θ + 25°) = cos 65 (Simplify your answer. Do not include the degree symbol in your answer.)

Acute Angles Write the following function in terms of its cofunction. Assume that all angles in which an unknown appears are acute angles. sin (θ + 25°) ___ sin (θ + 25°) = cos 65 (Simplify your answer. Do not include the degree symbol in your answer.)

Answer: \( \cos(65^\circ - \theta) \)

Explanation: The problem involves expressing the sine function in terms of its cofunction, cosine. The cofunction identity for sine and cosine states that \(\sin(90^\circ - x) = \cos(x)\). We need to express \(\sin(\theta + 25^\circ)\) using this identity.

Steps:

  1. Identify the Cofunction Identity:
  • The cofunction identity for sine is \(\sin(90^\circ - x) = \cos(x)\).
  1. Rewrite the Expression:
  • We have \(\sin(\theta + 25^\circ)\).
  • To use the cofunction identity, we need to express this in the form \(\sin(90^\circ - x)\).
  1. Set up the Equation:
  • Let \(x = 90^\circ - (\theta + 25^\circ)\).
  1. Simplify:

\[ x = 90^\circ - \theta - 25^\circ = 65^\circ - \theta \]

  1. Apply the Cofunction Identity:
  • Using \(\sin(90^\circ - x) = \cos(x)\), we have:

\[ \sin(\theta + 25^\circ) = \cos(65^\circ - \theta) \]

Therefore, \(\sin(\theta + 25^\circ)\) is expressed as \(\cos(65^\circ - \theta)\).

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