Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number (or angle) identity to find an expression for the exact value for each trigonometric function. $$\cos 67.5^{\circ}$$

Step-by-Step Solution

Answer

\(\cos(67.5^\circ) = \frac{\sqrt{2 - \sqrt{2}}}{2}\).

Step 1: Identify the Half-Angle Formula for Cosine

To find the cosine of \(67.5^\circ\), we need to use the half-angle formula for cosine. The half-angle identity is given by:\[ \cos\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 + \cos(A)}{2}} \] where the sign depends on the quadrant in which \(\frac{A}{2}\) is located.

Step 2: Establish the Initial Angle

Since we have \(67.5^\circ\), we notice that it is half of \(135^\circ\). Therefore, we can apply the half-angle formula where \(A = 135^\circ\).

Step 3: Calculate \(\cos(135^\circ)\)

We know \(135^\circ\) is in the second quadrant where cosine is negative. The reference angle for \(135^\circ\) is \(45^\circ\), and \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\). Therefore, \(\cos(135^\circ) = -\frac{\sqrt{2}}{2}\).

Step 4: Apply the Half-Angle Formula

Substitute \(\cos(135^\circ) = -\frac{\sqrt{2}}{2}\) into the half-angle formula:\[\cos(67.5^\circ) = \pm \sqrt{\frac{1 + \left(-\frac{\sqrt{2}}{2}\right)}{2}}\]Simplifying the expression inside the square root yields: \[\sqrt{\frac{2 - \sqrt{2}}{4}}\].

Step 5: Simplify the Expression

Continuing from the previous step, we simplify the square root expression:\[\cos(67.5^\circ) = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}\] Since \(67.5^\circ\) is in the first quadrant where cosine is positive, we choose the positive square root.

Key Concepts

Half-Angle Identities

Half-angle identities are crucial in trigonometry because they allow us to find the trigonometric function values for angles that are not commonly found on the unit circle. Essentially, they come in handy when dealing with angles that can be written as half of more familiar angles.
For the cosine function, the half-angle identity is given by:

• \( \cos\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 + \cos(A)}{2}} \)

The formula tells us how to find the cosine of an angle that is half of a given angle \( A \). The cool part about these identities is that they use already known values, making calculations easier.
However, we must consider the sign of the result, which depends on the quadrant of the angle \( \frac{A}{2} \). In the example exercise, \( 67.5^\circ \) being in the first quadrant indicates that we take the positive result.

Cosine Function

The cosine function is one of the basic trigonometric functions, often abbreviated as cos. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. The function is an essential tool in geometry, physics, and engineering.
In trigonometry, the cosine function also describes the „x-coordinate" on the unit circle, depending on an angle's rotation from the positive x-axis.

• For example, \( \cos(0^\circ) = 1 \) because it's aligned with the positive x-axis.
• \( \cos(90^\circ) = 0 \) due to its alignment with the y-axis.

In the step-by-step exercise, we've focused on using the cosine of \( 135^\circ \). This angle is in the second quadrant where cosine values are fundamentally negative. Like many trigonometric values, \( \cos(135^\circ) \) is evaluated using its reference angle \( 45^\circ \), modifying its sign accordingly.

Quadrant Angles

Understanding quadrant angles is key to determining the sign of trigonometric functions, including the cosine. The coordinate plane is divided into four quadrants:

• First Quadrant: where both sine and cosine are positive.
• Second Quadrant: where sine is positive, but cosine is negative.
• Third Quadrant: where both sine and cosine are negative.
• Fourth Quadrant: where sine is negative, but cosine is positive.

Knowing which quadrant an angle lies in allows us to decide between the positive or negative solution of trigonometric identities.
In our exercise, since \( 67.5^\circ \) is in the first quadrant, the cosine is positive. However, \( 135^\circ \) is in the second quadrant where cosine values are always negative. This distinction is vital for applying the correct sign when using half-angle identities.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, providing a link between angles and side lengths. They include sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. These functions model periodic phenomena such as sound and light waves.
Here’s a brief overview:

• Sine (sin): Opposite over hypotenuse in a right-angled triangle.
• Cosine (cos): Adjacent over hypotenuse.
• Tangent (tan): Opposite over adjacent.

Each function also has identities and relationships. These identities, like the half-angle identities used in the example, are essential for simplifying and solving trigonometric problems.
Understanding these functions and their identities allows you to accurately model and solve real-world and abstract mathematical problems efficiently.