Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Use half-angle formulas to find the exact values. (a) \(\cos 67^{\circ} 30^{\prime}\) (b) \(\sin 15^{\circ}\) (c) \(\tan \frac{3 \pi}{8}\)

Step-by-Step Solution

Answer

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

Step 1: Understand Half-Angle Formulas

The half-angle formulas are used for determining the exact values of trigonometric functions for angles that are not typically found on the unit circle. The half-angle formula for cosine is \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \), for sine is \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \), and for tangent is \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \). The sign depends on the quadrant that \( \frac{\theta}{2} \) is in.

Step 2: Convert Degrees and Minutes

Convert the angle \( 67^{\circ} 30^{\prime} \) into decimal degrees for easier calculation. There are 60 minutes in a degree, so \( 67^{\circ} 30^{\prime} = 67.5^{\circ} \).

Step 3: Apply Half-Angle Formula for Cosine

Use the half-angle formula for cosine on \( 67.5^{\circ} \) knowing it is \( \frac{135^{\circ}}{2} \). First, find \( \cos 135^{\circ} = -\frac{\sqrt{2}}{2} \).Then: \[ \cos 67.5^{\circ} = \sqrt{\frac{1 + \cos(135^{\circ})}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

Step 4: Apply Half-Angle Formula for Sine

Use the half-angle formula for sine on \( 15^{\circ} \) knowing it is \( \frac{30^{\circ}}{2} \). First, find \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).Then:\[ \sin 15^{\circ} = \sqrt{\frac{1 - \cos(30^{\circ})}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \]

Step 5: Apply Half-Angle Formula for Tangent

Use the half-angle formula for tangent on \( \frac{3\pi}{8} \) knowing it is \( \frac{3\pi/4}{2} \). First, find \( \tan \frac{3\pi}{4} = -1 \).Then:\[ \tan \frac{3\pi}{8} = \sqrt{\frac{1 - \cos(\frac{3\pi}{4})}{1 + \cos(\frac{3\pi}{4})}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 - \frac{\sqrt{2}}{2}}} \]This simplifies to \( 2 + \sqrt{2} \) the tangent half-angle identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. They form the backbone of trigonometry and allow us to simplify complex expressions and solve equations. One of the key types of these identities is the half-angle formulas. These are specifically useful when dealing with angles that are not standard angles of the unit circle. The half-angle formulas are given by:

• For cosine: \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)
• For sine: \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• For tangent: \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \)

The sign of each formula depends on the quadrant in which \( \frac{\theta}{2} \) lies, dictating whether the sine, cosine, or tangent of the angle takes a positive or negative value. Understanding these identities helps in breaking complex angles into manageable calculations and obtaining exact values of trigonometric functions.

Angle Conversion

When solving trigonometry problems, converting angles from degrees to radians, or vice versa, is often required. The conversion between degrees and radians is an essential skill.

• 1 degree equals \( \frac{\pi}{180} \) radians.
• Conversely, 1 radian equals \( \frac{180}{\pi} \) degrees.

In the provided exercise, one example of angle conversion was taking \( 67^{\circ} 30^{\prime} \) and converting it to decimal degrees as \( 67.5^{\circ} \). This standardization of the angle makes it easier to apply the half-angle formulas, as mathematical operations tend to be simpler in a single format.
Another aspect is knowing how to "translate" between special angles (like \( 30^{\circ}, 45^{\circ}, \) etc.) and radians, helping in solving problems that involve radian measures such as \( \frac{3\pi}{8} \). Mastery of these conversions is crucial for tackling problems across different contexts in trigonometry.

Exact Trigonometric Values

Exact trigonometric values refer to the precise values of trigonometric functions for certain angles. These are incredibly useful as they allow us to find exact solutions rather than approximate them with decimals. They are defined for common angles such as \( 30^{\circ}, 45^{\circ}, 60^{\circ} \), and their equivalents in radians.
A key advantage of using half-angle formulas is that they provide pathways to derive exact values for angles that are not typically standard. For instance:

• \( \cos 67.5^{\circ} \) derives from \( 135^{\circ} \) and uses standard values from \( \cos 135^{\circ} \).
• Similarly, \( \sin 15^{\circ} \) derives from \( 30^{\circ} \) with you knowing \( \cos 30^{\circ} \).
• Additionally, \( \tan \frac{3\pi}{8} \) derives its solution using standard radian calculations.

Access to these exact values through half-angle formulas simplifies complex calculations and enhances understanding of trigonometric principles, making it easier to work across a variety of math problems.