Understanding the Problem: Calculate sin 7α - cos 7β

The question asks us to find the value of the expression sin 7α - cos 7β given that both α and β are equal to 15°. We need to substitute these values and then evaluate the trigonometric expression.

Step-by-Step Calculation of 7α and 7β

We are given:

• α = 15°
• β = 15°

First, let's calculate the values of 7α and 7β:

7α = 7 × 15° = 105°

7β = 7 × 15° = 105°

So, the expression becomes sin 105° - cos 105°.

Evaluating sin 105° and cos 105°

To find the values of sin 105° and cos 105°, we can use trigonometric sum formulas. We can write 105° as the sum of two standard angles, for example, 60° + 45°.

Using the sum formula for sine, sin(A + B) = sin A cos B + cos A sin B:

sin 105° = sin(60° + 45°)

sin 105° = sin 60° cos 45° + cos 60° sin 45°

\(\sin 105^\circ = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right)\)

\(\sin 105^\circ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}}\)

Using the sum formula for cosine, cos(A + B) = cos A cos B - sin A sin B:

cos 105° = cos(60° + 45°)

cos 105° = cos 60° cos 45° - sin 60° sin 45°

\(\cos 105^\circ = \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right)\)

\(\cos 105^\circ = \frac{1}{2\sqrt{2}} - \frac{\sqrt{3}}{2\sqrt{2}} = \frac{1 - \sqrt{3}}{2\sqrt{2}}\)

Calculating sin 7α - cos 7β

Now, substitute the calculated values back into the expression sin 105° - cos 105°:

\(\sin 105^\circ - \cos 105^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} - \frac{1 - \sqrt{3}}{2\sqrt{2}}\)

Combine the fractions:

\(= \frac{(\sqrt{3} + 1) - (1 - \sqrt{3})}{2\sqrt{2}}\)

Remove the parentheses in the numerator:

\(= \frac{\sqrt{3} + 1 - 1 + \sqrt{3}}{2\sqrt{2}}\)

Simplify the numerator:

\(= \frac{2\sqrt{3}}{2\sqrt{2}}\)

Cancel out the common factor of 2:

\(= \frac{\sqrt{3}}{\sqrt{2}}\)

The value of sin 7α - cos 7β is \(\frac{\sqrt{3}}{\sqrt{2}}\).

Final Answer Summary

Given α = β = 15°, we found that 7α = 105° and 7β = 105°. The expression is sin 105° - cos 105°.

We calculated sin 105° = \(\frac{\sqrt{3} + 1}{2\sqrt{2}}\) and cos 105° = \(\frac{1 - \sqrt{3}}{2\sqrt{2}}\).

Subtracting these values, we got sin 105° - cos 105° = \(\frac{\sqrt{3}}{\sqrt{2}}\).

This matches one of the given options.