Questions: Find the exact value of cos(75°)

Solution Steps

To find the exact value of \(\cos(75^\circ)\), we can use the angle addition formula for cosine. Specifically, we can express \(75^\circ\) as the sum of two angles whose cosine and sine values are known, such as \(45^\circ\) and \(30^\circ\). The formula is:

\[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \]

Using this formula, we can calculate \(\cos(75^\circ)\) by substituting \(a = 45^\circ\) and \(b = 30^\circ\).

To find \(\cos(75^\circ)\), we can use the angle addition formula: \[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \] where we let \(a = 45^\circ\) and \(b = 30^\circ\).

We calculate the trigonometric values for \(a\) and \(b\):

• \(\cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071\)
• \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660\)
• \(\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071\)
• \(\sin(30^\circ) = \frac{1}{2} = 0.5000\)

Substituting these values into the angle addition formula: \[ \cos(75^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \] \[ \cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \] \[ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \]

Final Answer