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Chapter 5: Problem 34

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$67^{\circ} 30^{\prime}$$

Short Answer

Expert verified
The exact values of sine, cosine, and tangent for the angle are given by \( \sin(67.5^{\circ}) = \frac{ \sqrt{2 + \sqrt{2}} }{2}\), \( \cos(67.5^{\circ}) = \frac{ \sqrt{2 - \sqrt{2}} }{2}\), and \( \tan(67.5^{\circ}) = 1 + \sqrt{2}\), respectively.

Step by step solution

01

Convert the given angle to degrees.

Here, the given angle is \(67^{\circ} 30^{\prime}\). Convert it to degrees: \(67^{\circ} 30^{\prime} = 67.5^{\circ}\).
02

Double the angle.

Double the angle so that it aligns with a well-known trigonometric value. This gives \(2 \times 67.5^{\circ} = 135^{\circ}\). Using the unit circle, we know that \(\cos(135^{\circ}) = -\frac{1}{\sqrt{2}}\).
03

Apply the half-angle formulas.

Plug these values into the half-angle formulas to find the sine, cosine, and tangent values: \( \sin(67.5^{\circ}) = \sqrt{\frac{1 - (-\frac{1}{\sqrt{2}})}{2}}\), \( \cos(67.5^{\circ}) = \sqrt{\frac{1 + (-\frac{1}{\sqrt{2}})}{2}}\), \( \tan(67.5^{\circ}) = \frac{\sin(67.5^{\circ})}{\cos(67.5^{\circ})}\). Simplify each quantity to obtain the final values.

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