Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Step-by-Step Solution

Answer

The exact values are \(\sin(\frac{13 \pi}{12}) = \frac{\sqrt{2} + \sqrt{6}}{4}\), \(\cos(\frac{13 \pi}{12}) = \frac{\sqrt{2} - \sqrt{6}}{4}\) and \(\tan(\frac{13 \pi}{12}) = -\sqrt{3}\)

Step 1: Split the angle

First, the angle \(\frac{13 \pi}{12}\) should be expressed as a combination of common angles which we already know the sine, cosine and tangent values of. The most reasonable way would be to represent \(\frac{13 \pi}{12}\) as \(\frac{\pi}{4} + \frac{\pi}{3}\), because it’s easy to remember the sine, cosine and tangent of these angles, which are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{3}}{2}\) and \(1\), \( \frac{\sqrt{3}}{2}\), respectively.

Step 2: Apply sum and difference identities

With the new split, apply the sum and difference identities:The sine of sum of two angles \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\)The cosine of sum of two angles \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\) Therefore, \(\sin(\frac{13 \pi}{12}) = \sin(\frac{\pi}{4} + \frac{\pi}{3}) = \sin(\frac{\pi}{4})\cos(\frac{\pi}{3}) + \cos(\frac{\pi}{4})\sin(\frac{\pi}{3})\) and \(\cos(\frac{13 \pi}{12}) = \cos(\frac{\pi}{4} + \frac{\pi}{3}) = \cos(\frac{\pi}{4})\cos(\frac{\pi}{3}) - \sin(\frac{\pi}{4})\sin(\frac{\pi}{3})\)

Step 3: Calculation and Simplification

Substitute the known values in: \(\sin(\frac{13 \pi}{12}) = \frac{\sqrt{2}}{2} \times \frac{1}{2} + \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}\)\(\cos(\frac{13 \pi}{12}) = \frac{\sqrt{2}}{2} \times \frac{1}{2} - \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4}\)Then, calculate the tangent by dividing the sine of the angle by its cosine:\(\tan(\frac{13 \pi}{12}) = \frac{\sin(\frac{13 \pi}{12})}{\cos(\frac{13 \pi}{12}}) = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{\frac{\sqrt{2} - \sqrt{6}}{4}} = -\sqrt{3}\)

Step 4: Rationalize the denominator

For the tangential result, rationalize the denominator by multiplying and dividing by the conjugate \((\sqrt{2} + \sqrt{6})\). Then the tangent is \(-\sqrt{3}\)