Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Use half-angle formulas to find the exact values. (a) \(\cos 165^{\circ}\) (b) \(\sin 157^{\circ} 30^{\prime}\) (c) \(\tan \frac{\pi}{8}\)

Step-by-Step Solution

Answer

(a) \( -\sqrt{\frac{2 + \sqrt{3}}{2}} \), (b) \( -\sqrt{\frac{2 - \sqrt{2}}{2}} \), (c) \( \sqrt{2} - 1 \)

Step 1: Express angle as half-angles

To use the half-angle formulas, we first express the given angles as half of a known angle. For (a), 165° can be expressed as 330°/2. For (b), 157° 30′ can be expressed as 315°/2. For (c), since \( \frac{\pi}{8} = \frac{\pi/4}{2} \), it's already in a half-angle form.

Step 2: Half-angle formula for cosine

Using the half-angle formula \( \cos \theta = \pm \sqrt{\frac{1 + \cos(2\theta)}{2}} \), we substitute \( 2\theta = 330^{\circ} \) for part (a). This gives the expression \( \cos 165^{\circ} = -\sqrt{\frac{1 + \cos(330^{\circ})}{2}} \). Since \( \cos(330^{\circ}) = \cos(360^{\circ} - 30^{\circ}) = \cos(30^{\circ}) = \sqrt{3}/2 \), this simplifies to \( \cos 165^{\circ} = -\sqrt{\frac{1 + \sqrt{3}/2}{2}} \).

Step 3: Half-angle formula for sine

Using the half-angle formula \( \sin \theta = \pm \sqrt{\frac{1 - \cos(2\theta)}{2}} \), substitute \( 2\theta = 315^{\circ} \) for part (b). This gives us \( \sin 157^{\circ} 30^{\prime} = -\sqrt{\frac{1 - \cos(315^{\circ})}{2}} \). Since \( \cos(315^{\circ}) = \cos(360^{\circ} - 45^{\circ}) = \cos(45^{\circ}) = \sqrt{2}/2 \), we have \( \sin 157^{\circ} 30^{\prime} = -\sqrt{\frac{1 - \sqrt{2}/2}{2}} \).

Step 4: Half-angle formula for tangent

Using the half-angle formula \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \) for part (c). Here, \( \theta = \frac{\pi/4} \), so \( \tan \frac{\pi}{8} = \pm \sqrt{\frac{1 - \cos(\pi/4)}{1 + \cos(\pi/4)}} \). Given \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \), this simplifies to \( \tan \frac{\pi}{8} = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} \), which further simplifies using rationalization techniques.

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions. Think of them as "tools" that help you simplify and solve complex trigonometric equations. When dealing with angles that are not straightforward, like in this exercise, identities allow us to break them down into more manageable parts. This is particularly useful when trying to find exact values without a calculator. Some common trigonometric identities include the Pythagorean identities, angle sum and difference identities, and half-angle identities. The half-angle identities specifically,

• \( \cos \theta = \pm \sqrt{\frac{1 + \cos(2\theta)}{2}} \)
• \( \sin \theta = \pm \sqrt{\frac{1 - \cos(2\theta)}{2}} \)
• \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \)

allow us to find trigonometric values for half of a given angle. Remember, the sign ahead of the square root depends on the quadrant in which the angle \(\theta\) lies.

Angle Conversion Techniques

Converting and manipulating angles is often necessary when solving trigonometric problems. In our example, we need to convert angles to be in a half-angle form. This involves expressing complex angle measurements in a way that they are half of a familiar or standard angle. For part (a), we express 165° as \(330°/2\), which makes it ready for the cosine half-angle formula. Similarly, in part (b), the angle 157° 30′ can be written as 315°/2, making it suitable for the sine half-angle formula. Lastly, part (c) requires no conversion since \(\frac{\pi}{8} = \frac{\pi/4}{2}\) is already in the format we need. Notice how having a grasp of basic angle conversions can simplify the process of applying trigonometric identities. Often these conversions aren’t straightforward, but with practice they become easier to recognize.

Finding Exact Values in Trigonometry

Finding exact values means discovering the precise trigonometric value of an angle without resorting to a calculator. This is done by leveraging known trigonometric values of standard angles, such as \(0°, 30°, 45°, 60°, 90°,\) and their radian equivalents. Applying the half-angle formulas effectively yields these exact values.

• For example, to find \(\cos 165°\), we use the cosine half-angle identity with \(330°\), relying on the known value \(\cos(30°) = \sqrt{3}/2\).
• Finding \(\sin 157° 30'\) involves knowing \(\cos(45°) = \sqrt{2}/2\) and substituting into the sine half-angle formula.
• To determine \(\tan \frac{\pi}{8}\), understanding the exact value \(\cos(\pi/4) = \sqrt{2}/2\) aids in simplifying the tangent half-angle formula.

Being comfortable with these computations results in precise results. Not only does this deepen your understanding of trigonometry, but also boosts problem-solving efficiency.