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Chapter 5: Problem 61

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$112^{\circ} 30^{\prime}$$

Short Answer

Expert verified
The exact values of the sine, cosine, and tangent of the angle \(112.5^{\circ}\) are \(\sqrt{\frac{1 - \sqrt{2}}{4}}\), \(\sqrt{\frac{1 + \sqrt{2}}{4}}\), and \(\frac{1 - \sqrt{2}}{1 + \sqrt{2}}\) respectively.

Step by step solution

01

Convert Angle to Decimal Form

Convert \(112^{\circ} 30^{\prime}\) to decimal form. One minute is equal to \(\frac{1}{60}\) degrees, so \(30\) minutes is equal to \(\frac{30}{60} = 0.5\) degrees. Add this to the degrees part to get the full angle in decimal form: \(112^{\circ} + 0.5^{\circ} = 112.5^{\circ}\)
02

Convert Degrees to Radians

You need to convert this angle from degrees to radians, because the trigonometric functions in the half-angle formulas work with radians. The conversion formula is \( \theta (in \, radians) = \theta (in \, degrees) \times \frac{\pi}{180}\). Therefore, \(112.5^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{8}\)
03

Apply the Half-Angle Formulas

Plug \(\frac{5\pi}{8}\) into the half-angle formulas to calculate \(\sin\frac{5\pi}{16}\), \(\cos\frac{5\pi}{16}\), and \(\tan\frac{5\pi}{16}\). Also note that because the angle \(\frac{5\pi}{16}\) is in the first quadrant (where both sine and cosine are positive), we can drop the \(\pm\) in the formulas.
04

Calculate The Sine Value

\(\sin\frac{5\pi}{16} = \sqrt{\frac{1 - \cos\frac{5\pi}{8}}{2}} = \sqrt{\frac{1 - \cos\frac{5\pi}{4}}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{1 - \sqrt{2}}{4}}\)
05

Calculate The Cosine Value

\(\cos\frac{5\pi}{16} = \sqrt{\frac{1 + \cos\frac{5\pi}{8}}{2}} = \sqrt{\frac{1 + \cos\frac{5\pi}{4}}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{1 + \sqrt{2}}{4}}\)
06

Calculate The Tangent Value

\(\tan\frac{5\pi}{16} = \sqrt{\frac{1 - \cos\frac{5\pi}{8}}{1 + \cos\frac{5\pi}{8}}} = \sqrt{\frac{1 - \cos\frac{5\pi}{4}}{1 + \cos\frac{5\pi}{4}}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} = \frac{1 - \sqrt{2}}{1 + \sqrt{2}}\)

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