Use 105° = 135° - 30° to find the exact value of 105°.

Verified step by step guidance

1. Recognize that 105° can be expressed as the difference of two angles: 135° and 30°.
2. Use the angle subtraction identity for cosine: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
3. Substitute \( a = 135° \) and \( b = 30° \) into the identity: \( \cos(105°) = \cos(135°)\cos(30°) + \sin(135°)\sin(30°) \).
4. Recall the exact trigonometric values: \( \cos(135°) = -\frac{\sqrt{2}}{2} \), \( \sin(135°) = \frac{\sqrt{2}}{2} \), \( \cos(30°) = \frac{\sqrt{3}}{2} \), and \( \sin(30°) = \frac{1}{2} \).
5. Substitute these values into the expression and simplify to find the exact value of \( \cos(105°) \).

Key Concepts

Angle Addition and Subtraction

The angle addition and subtraction formulas are fundamental in trigonometry, allowing us to express the sine, cosine, and tangent of sums or differences of angles. For example, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b) helps in calculating the trigonometric values of angles that are not standard. Understanding these formulas is essential for breaking down complex angles into manageable components.

Reference Angles

Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are crucial for determining the sine, cosine, and tangent values of angles in different quadrants. For instance, the reference angle for 105° is 75°, which helps in finding the exact trigonometric values by relating them to known angles.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often derived from special triangles or the unit circle. For example, knowing that sin(30°) = 1/2 and cos(30°) = √3/2 allows us to compute values for angles like 105° using angle addition or subtraction. Mastery of these exact values is essential for solving trigonometric equations and problems.