Match each expression in Column I with its value in Column II.
10. cos 67.5°

Verified step by step guidance

1. Recognize that the angle 67.5° is a special angle that can be expressed as the sum or difference of two common angles. Specifically, 67.5° = 45° + 22.5° or 67.5° = 90° - 22.5°.
2. Use the cosine addition formula if you choose to express 67.5° as 45° + 22.5°. The formula is: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
3. Alternatively, use the complementary angle identity \(\cos(90^\circ - \theta) = \sin \theta\) to rewrite \(\cos 67.5^\circ\) as \(\sin 22.5^\circ\).
4. Recall or derive the exact values of \(\sin 22.5^\circ\) or \(\cos 22.5^\circ\) using half-angle formulas, for example: \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}\).
5. Substitute the known values into the chosen formula to express \(\cos 67.5^\circ\) in terms of square roots and fractions, which can then be matched to the corresponding value in Column II.

Key Concepts

Cosine of Special Angles

Certain angles like 30°, 45°, 60°, and their multiples have known exact cosine values. Understanding these helps in evaluating expressions like cos 67.5°, which can be derived from these special angles using angle formulas.

Half-Angle Formula

The half-angle formula expresses the cosine of half an angle in terms of the cosine of the original angle: cos(θ/2) = ±√((1 + cos θ)/2). This formula is essential for finding cos 67.5°, since 67.5° = 135°/2.

Sign Determination in Trigonometric Values

When using formulas involving square roots, determining the correct sign (+ or -) is crucial. For angles in the first quadrant (0° to 90°), cosine values are positive, guiding the correct choice of sign in calculations.