In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.cos(45° + 30°)

Verified step by step guidance

1. Identify the sum formula for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
2. Substitute \( a = 45^\circ \) and \( b = 30^\circ \) into the formula.
3. Calculate \( \cos 45^\circ \) and \( \cos 30^\circ \).
4. Calculate \( \sin 45^\circ \) and \( \sin 30^\circ \).
5. Substitute these values into the formula: \( \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \).

Key Concepts

Sum and Difference Formulas

Sum and difference formulas are trigonometric identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, the cosine of the sum of two angles is given by cos(A + B) = cos(A)cos(B) - sin(A)sin(B). These formulas are essential for simplifying expressions involving the addition or subtraction of angles.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values allows for quick calculations without the need for a calculator. For instance, cos(45°) = √2/2 and cos(30°) = √3/2, which are crucial for applying the sum formula in this problem.

Angle Measurement in Degrees

Angle measurement in degrees is a way of quantifying angles, where a full circle is divided into 360 equal parts. In trigonometry, angles can be expressed in degrees or radians, but for this problem, we are using degrees. Understanding how to convert between degrees and radians, as well as how to visualize angles on the unit circle, is important for applying trigonometric identities effectively.