Questions: Identify the exact value of the following expression: cos(75°) √6 + √2 / 4 √6 - √2 / 4 -√6 + √2 / 4 -√6 - √2 / 4 None of these are correct.

Solution Steps

To find the exact value of \(\cos(75^\circ)\), we can use the angle addition formula for cosine: \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). We can express \(75^\circ\) as \(45^\circ + 30^\circ\) and then use the known values of \(\cos(45^\circ)\), \(\sin(45^\circ)\), \(\cos(30^\circ)\), and \(\sin(30^\circ)\).

1. Use the angle addition formula: \(\cos(75^\circ) = \cos(45^\circ + 30^\circ)\).
2. Substitute the known values: \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), and \(\sin(30^\circ) = \frac{1}{2}\).
3. Simplify the expression to find the exact value.

To find \(\cos(75^\circ)\), we use the angle addition formula: \[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \]

We substitute the known values: \[ \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071 \] \[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071 \] \[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \] \[ \sin(30^\circ) = \frac{1}{2} = 0.5000 \]

Substitute these values into the formula: \[ \cos(75^\circ) = (0.7071 \times 0.8660) - (0.7071 \times 0.5000) \] \[ \cos(75^\circ) = 0.6124 - 0.3536 \] \[ \cos(75^\circ) \approx 0.2588 \]

The calculated value of \(\cos(75^\circ)\) is approximately \(0.2588\). We compare this with the given options: \[ \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659 \] \[ \frac{\sqrt{6}-\sqrt{2}}{4} \approx 0.2588 \] \[ \frac{-\sqrt{6}+\sqrt{2}}{4} \approx -0.2588 \] \[ \frac{-\sqrt{6}-\sqrt{2}}{4} \approx -0.9659 \]

The value \(\frac{\sqrt{6}-\sqrt{2}}{4}\) matches our calculated value of \(\cos(75^\circ)\).

Final Answer