Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$-\frac{7 \pi}{12}$$

Step-by-Step Solution

Answer

\(\sin(-\frac{7 \pi}{12}) = \sqrt{2 - \sqrt{3}}\), \(\cos(-\frac{7 \pi}{12}) = -1 + \sqrt{3}\), \(\tan(-\frac{7 \pi}{12}) = 2 - \sqrt{3}\)

Step 1: Express the given angle as the sum or difference of two angles

The angle \(-\frac{7 \pi}{12}\) can be expressed as the sum of \(-\frac{\pi}{4}\) and \(-\frac{\pi}{3}\). Check this by adding the two angles: \(-\frac{\pi}{4} + -\frac{\pi}{3} = -\frac{7 \pi}{12}\).

Step 2: Use sum or difference identities to find sine, cosine, and tangent

Using sum and difference identities, the trigonometric functions can be calculated directly. The sine, cosine, and tangent of \(-\frac{\pi}{4}\) and \(-\frac{\pi}{3}\) are known values. Substitute those values into equations:\n Sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\),\n Cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\),\n Tangent: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Then, the exact values of the sine, cosine, and tangent of \(-\frac{7 \pi}{12}\) can be calculated.

Step 3: Calculate the exact values of the sine, cosine, and tangent

Substituting the known values into the above equations, the exact values are calculated as follows :\n \(\sin(-\frac{7 \pi}{12}) = \sqrt{2 - \sqrt{3}}\), \n \(\cos(-\frac{7 \pi}{12}) = -1 + \sqrt{3}\), \n \(\tan(-\frac{7 \pi}{12}) = 2 - \sqrt{3}\).