Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$2 \cos ^{2} 67.5^{\circ}-1$$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{\sqrt{2}}{2}\).

Step 1: Identify the Identity

The expression \(2 \cos^2 \theta - 1\) is a well-known trigonometric identity, which can be simplified using the double angle identity: \(2 \cos^2 \theta - 1 = \cos 2\theta\). Our goal is to express the given expression as a single trigonometric function or a number.

Step 2: Substitute the Angle

Substitute \(\theta = 67.5^\circ\) into the expression for the cosine double angle identity. This gives us \(\cos(2 \times 67.5^\circ)\).

Step 3: Calculate the Double Angle

Calculate the angle for the double angle identity. \(2 \times 67.5^\circ = 135^\circ\). So, the expression simplifies to \(\cos 135^\circ\).

Step 4: Evaluate \(\cos 135^\circ\) in Exact Form

Determine \(\cos 135^\circ\) using the unit circle. \(135^\circ\) is in the second quadrant, where cosine is negative. By symmetry, \(\cos 135^\circ = -\cos 45^\circ\). \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), therefore, \(\cos 135^\circ = -\frac{\sqrt{2}}{2}\).

Key Concepts

Double Angle Formula

The Double Angle Formula is an essential tool in trigonometry that simplifies expressions and calculations. It provides a way to relate angles that are multiples of each other, making complex trigonometric problems more manageable.
One of the most common forms of this formula is for the cosine function:

• \( \cos 2\theta = 2 \cos^2 \theta - 1 \)
• Alternatively, \( \cos 2\theta = 1 - 2 \sin^2 \theta \)
• Or \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

These variations of the formula help in different scenarios, depending on which trigonometric functions are known or required. In the context of the original exercise, we used \( 2 \cos^2 \theta - 1 = \cos 2\theta \), allowing us to transform a more complex expression into a simpler function of the angle double the original.

Cosine Function

The cosine function is a fundamental part of trigonometry and represents the horizontal coordinate of a point on the unit circle as it traverses around the circle. Cosine is one of the core trigonometric functions, alongside sine and tangent.
Some key characteristics of the cosine function include:

• Periodicity: Cosine is a periodic function with a period of \(360^\circ\) or \(2\pi\). That means \(\cos(\theta) = \cos(\theta + 360^\circ)\).
• Symmetry: Cosine is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• Amplitude: The range of cosine is from -1 to 1, which corresponds to its amplitude in the unit circle context.

In the solution, the evaluation of \(\cos 135^\circ\) relies on recognizing its position on the unit circle, using the property that cosine is negative in the second quadrant.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It is a powerful tool in trigonometry, used to define the sine, cosine, and tangent functions for all real numbers.
Here's how the unit circle is utilized:

• Every point \((x, y)\) on the unit circle corresponds to a cosine and sine value: \(\cos\theta = x\) and \(\sin\theta = y\).
• The unit circle helps visualize angles and their corresponding trig values. For example, \(135^\circ\) is located in the second quadrant where cosine values are negative.
• Symmetry and special angles assist in the quick calculation of values. For instance, knowing \(45^\circ\) has \(\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}\), we can deduce that \(\cos 135^\circ = -\cos 45^\circ\).

Thus, not only does the unit circle offer a visual representation of trigonometric principles, but it also provides an effective method to calculate specific angle values accurately.