Problem 20 - Algebra and Trigonometry

Question

15–26 Use an appropriate half-angle formula to find the exact value of the expression. $$\cos 112.5^{\circ}$$

Step-by-Step Solution

Answer

\( \cos 112.5^{\circ} = -\frac{\sqrt{2 - \sqrt{2}}}{2} \).

Step 1: Identify Half-Angle Formula for Cosine

The half-angle formula for cosine is \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \). We will use this formula to find \( \cos 112.5^{\circ} \).

Step 2: Find Double Angle

Since \( 112.5^{\circ} \) is half of \( 225^{\circ} \), set \( x = 225^{\circ} \). This choice lets us use the half-angle formula with \( x \).

Step 3: Calculate \( \cos 225^{\circ} \)

The cosine of \( 225^{\circ} \) can be calculated as it is in the third quadrant where both sine and cosine are negative. Therefore, \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \).

Step 4: Plug into Half-Angle Formula

Substitute \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \) into the half-angle formula: \( \cos 112.5^{\circ} = \pm \sqrt{\frac{1 + (-\frac{\sqrt{2}}{2})}{2}} \).

Step 5: Simplify the Expression

Simplify the expression: \( \cos 112.5^{\circ} = \pm \sqrt{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2} = \frac{2 - \sqrt{2}}{4} \), giving \( \cos 112.5^{\circ} = \pm \sqrt{\frac{2 - \sqrt{2}}{4}} \).

Step 6: Determine the Correct Sign

Since \( 112.5^{\circ} \) is in the second quadrant where cosine is negative, \( \cos 112.5^{\circ} = -\sqrt{\frac{2 - \sqrt{2}}{4}} \).

Step 7: Complete the Simplification

Simplifying further, \( \cos 112.5^{\circ} = -\frac{\sqrt{2 - \sqrt{2}}}{2} \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry, used to describe the relationship between the angles and sides of a right triangle. It is especially helpful when working with circular motion and oscillations. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This means that for any angle \( \theta \), \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

Cosine function values vary from -1 to 1, depending on the angle. Additionally, the cosine function is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians. This periodic nature makes it valuable in modeling situations that repeat at regular intervals.

A unique feature of the cosine function is its symmetry. It is an even function, meaning \( \cos(\theta) = \cos(-\theta) \). This characteristic helps in simplifying calculations in trigonometry.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are useful tools for simplifying expressions and solving trigonometric equations.

Some of the most common trigonometric identities include:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal Identities: \( \sin \theta = \frac{1}{\csc \theta} \), \( \cos \theta = \frac{1}{\sec \theta} \), \( \tan \theta = \frac{1}{\cot \theta} \)
• Sum and Difference Formulas:
  \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)\
• Double Angle and Half Angle: \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \)

These identities form the backbone of algebraic manipulation in trigonometry. In the context of this exercise, we used the half-angle formula for cosine to find the exact value of \( \cos 112.5^{\circ} \), by relating it to the known value of \( \cos 225^{\circ} \), which is part of the double angle approach.

Exact Trigonometric Values

Finding exact trigonometric values refers to the process of determining precise values, rather than approximations, for trigonometric functions at specific angles. These angles are usually multiples of \(30^{\circ}\), \(45^{\circ}\), or \(60^{\circ}\), for which values are typically memorized or derived quickly.

In our exercise, we calculated \( \cos 112.5^{\circ} \) using exact trigonometric steps:

• Recognized \( 112.5^{\circ} = \frac{225^{\circ}}{2} \)
• Calculated \( \cos 225^{\circ} = -\frac{\sqrt{2}}{2} \)
• Applied the half-angle formula \( \cos\left(\frac{225^{\circ}}{2}\right) \)
• Simplified to find \( \cos 112.5^{\circ} = -\frac{\sqrt{2 - \sqrt{2}}}{2} \)

This exact value shows how trigonometric identities, like the half-angle formula, allow us to compute these values accurately, avoiding the need for calculators and rounding. Mastery of exact trigonometric values enhances both understanding and confidence in tackling complex problems.