Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Step-by-Step Solution

Answer

The exact values based on the half-angle formulas are \[sin(75^{\circ})=\frac{\sqrt{4 - 2\sqrt{3}}}{2}\], \[cos(75^{\circ})=\frac{\sqrt{4 + 2\sqrt{3}}}{2}\] and \[tan(75^{\circ})= \sqrt{1 - \sqrt{3}}\]

Step 1: Calculate the Cosine

First, express the angle of 75 degrees as the sum of two known angles for which we can easily find the cosine. One combination is 45 and 30 degrees. Using the sum of angles identity for cosine, \[cos(x + y) = cos(x)cos(y) - sin(x)sin(y)\], we can calculate: \[cos(75) = cos(45 + 30) = cos(45)cos(30) - sin(45)sin(30) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\]

Step 2: Apply Half-Angle Formulas for Sine and Cosine

Next, we substitute the found cosine value into the half-angle formulas for sine and cosine: \[sin\left(\frac{75}{2}\right) = \sqrt{\frac{1 - cos(75)}{2}} = \sqrt{\frac{1 - (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} = \sqrt{\frac{2 - \sqrt{6} - sqrt{2}}{4}} = \frac{\sqrt{4 - 2\sqrt{3}}}{2}\], \[cos\left(\frac{75}{2}\right) = \sqrt{\frac{1 + cos(75)}{2}} = \sqrt{\frac{1 + (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} = \sqrt{\frac{2 + \sqrt{6} + sqrt{2}}{4}} = \frac{\sqrt{4 + 2\sqrt{3}}}{2}\]

Step 3: Apply Half-Angle Formula for Tangent

Lastly, substitute the found cosine value into the half-angle formula for the tangent: \[tan\left(\frac{75}{2}\right) = \sqrt{\frac{1 - cos(75)}{1 + cos(75)}} = \sqrt{\frac{1 - (\frac{\sqrt{6} + sqrt{2}}{4})}{1 + (\frac{\sqrt{6} + \sqrt{2}}{4})}} = \sqrt{1 - \sqrt{3}}\]