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Chapter 5: Problem 60

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}$$

Short Answer

Expert verified
The exact values of sine, cosine, and tangent for an angle of 165 degrees using half-angle formulas are as follows: cosine is negative (exact value to be determined by calculation), sine is positive (exact value to be determined by calculation), and tangent is positive (exact value also to be determined by calculation). These values are based on the knowledge that 165 degrees falls in the second quadrant where cosine is negative while sine and tangent are positive.

Step by step solution

01

Determine the Half-Angle

First, find the half angle of 165 degrees. This can be done by simply dividing the given angle by 2. So, \(\frac{165^{\circ}}{2} = 82.5^{\circ}\).
02

Use the Half-Angle Formulas

Next, apply the half-angle formulas. For cosine, the formula is \(\cos \frac{\theta}{2} = 卤 \sqrt{\frac{1+\cos \theta}{2}}\), for sine, the formula is \(\sin \frac{\theta}{2} = 卤 \sqrt{\frac{1-\cos \theta}{2}}\), and for tangent, the formula is \(\tan \frac{\theta}{2} = 卤 \sqrt{\frac{1-\cos \theta}{1+\cos \theta}}\). Note: The issue of the correct sign (卤) will depend on the quadrant of the given angle.
03

Calculate the Values

Now, substitute \(82.5^{\circ}\) into these formulas and calculate the values. Note that the cosine of 165 degrees will be negative, as it falls in the second quadrant. This implies a negative value for the cosine and positive values for the sine and tangent. So, we'll use the negative square root in the formula for the cosine and the positive square root for the sine and tangent.
04

Compute the Exact Values

Final step is to compute the exact values of cosine, sine, and tangent for the half angle. Hence, \(\cos 82.5^{\circ}= -\sqrt{\frac{1+\cos 165^{\circ}}{2}}\), \(\sin 82.5^{\circ}= \sqrt{\frac{1-\cos 165^{\circ}}{2}}\), and \(\tan 82.5^{\circ}= \sqrt{\frac{1-\cos 165^{\circ}}{1+\cos 165^{\circ}}}\). Using the unit circle or a scientific calculator, calculate these values.

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