Find the exact value of each expression. (Do not use a calculator.)
cos 105° (Hint: 105° = 60° + 45°)

Verified step by step guidance

1. Recognize that the problem involves finding the cosine of an angle that can be expressed as the sum of two known angles: 105° = 60° + 45°.
2. Use the cosine addition formula: $\cos (a+b)=\cos a\cos b-\sin a\sin b$.
3. Substitute $a=60°$ and $b=45°$ into the formula: $\cos (105°)=\cos (60°)\cos (45°)-\sin (60°)\sin (45°)$.
4. Recall the exact trigonometric values: $\cos (60°)=\frac{1}{2}$, $\cos (45°)=\frac{\sqrt{2}}{2}$, $\sin (60°)=\frac{\sqrt{3}}{2}$, and $\sin (45°)=\frac{\sqrt{2}}{2}$.
5. Substitute these values into the expression: $\cos (105°)=\left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)-\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)$.

Key Concepts

Cosine Addition Formula

The cosine addition formula states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B). This formula allows us to find the cosine of an angle that is the sum of two other angles, which is essential for calculating cos(105°) as 105° can be expressed as 60° + 45°.

Special Angles in Trigonometry

In trigonometry, certain angles like 30°, 45°, and 60° have known sine and cosine values. For example, cos(60°) = 1/2 and cos(45°) = √2/2. Recognizing these special angles helps simplify calculations and find exact values without a calculator.

Quadrants and Angle Signs

Understanding the unit circle and the signs of trigonometric functions in different quadrants is crucial. Since 105° is in the second quadrant, where cosine values are negative, this knowledge is important when determining the final value of cos(105°) after applying the cosine addition formula.