Use a calculator to evaluate each expression. cos 75°29' cos 14°31' - sin 75°29' sin 14°31'

Verified step by step guidance

1. Recognize that the expression \( \cos 75^\circ 29' \cos 14^\circ 31' - \sin 75^\circ 29' \sin 14^\circ 31' \) matches the cosine addition formula: \( \cos A \cos B - \sin A \sin B = \cos (A + B) \).
2. Rewrite the expression using the formula as \( \cos (75^\circ 29' + 14^\circ 31') \).
3. Add the angles: \( 75^\circ 29' + 14^\circ 31' \) by adding degrees and minutes separately.
4. Convert the resulting angle into decimal degrees if necessary for calculator input, remembering that 1 minute = \( \frac{1}{60} \) degrees.
5. Use a calculator to find \( \cos \) of the sum angle obtained in the previous step.

Key Concepts

Angle Measurement in Degrees and Minutes

Angles can be expressed in degrees (°) and minutes ('). One degree equals 60 minutes, so to work with these angles in calculations, convert minutes to decimal degrees by dividing by 60. For example, 75°29' equals 75 + 29/60 degrees.

Cosine of Sum of Angles Formula

The expression cos A cos B - sin A sin B is the cosine of the sum of two angles, i.e., cos(A + B). This identity allows simplification of the given expression by recognizing it as cos(75°29' + 14°31').

Using a Calculator for Trigonometric Values

Calculators can evaluate trigonometric functions when angles are in decimal degrees. After converting angles from degrees and minutes to decimal form, input the sum into the cosine function to find the numerical value accurately.