Problem 32 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\cos \frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt{2}}{2} \)

Step 1: Convert Angle to Degrees

Convert the angle from radians to degrees to understand its position better. Given angle: \( \frac{7\pi}{4} \).Multiply by \( \frac{180}{\pi} \) to convert: \[ \frac{7\pi}{4} \times \frac{180}{\pi} = 315^{\circ} \].

Step 2: Determine Reference Angle

Identify the reference angle for \(315^{\circ}\). This angle lies in the fourth quadrant.Subtract \(315^{\circ}\) from \(360^{\circ}\) (full circle):\[ 360^{\circ} - 315^{\circ} = 45^{\circ} \].The reference angle is \(45^{\circ}\).

Step 3: Use the Unit Circle

The cosine of the reference angle \(45^{\circ}\) is \(\frac{\sqrt{2}}{2}\).In the fourth quadrant, the cosine is positive, thus:\[ \cos 315^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \].

Step 4: Final Value

Thus, the exact value of \( \cos \frac{7 \pi}{4} \) is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Cosine

The cosine function is a fundamental part of trigonometry, representing the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function is often abbreviated to 'cos'. It is one of the primary trigonometric functions, along with sine and tangent. In the unit circle context, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Cosine has a range of values between -1 and 1 and is periodic with a period of 360 degrees or \(2\pi\) radians. This periodic nature means that the function repeats its values in regular intervals. Cosine is particularly useful because it helps describe the wave-like nature of periodic phenomena such as sound and light.

• **Form**: For any angle \(\theta\), \(\cos(\theta)\) gives the horizontal coordinate on the unit circle.
• **Positive in Quadrants**: Cosine is positive in the first and fourth quadrants, which is important for determining the sign of the angle's cosine value.

Angle Conversion

Angles can be expressed in two primary units: degrees and radians. Understanding angle conversion is crucial for working out trigonometric functions like cosine.

Degrees are more commonly used in everyday contexts, while radians are often used in calculus and higher mathematics because they make some formulas easier and more intuitive. For conversion:

• **Radians to Degrees**: Multiply the radian measure by \( \frac{180}{\pi} \) to convert to degrees.
• **Degrees to Radians**: Multiply the degree measure by \( \frac{\pi}{180} \) to convert to radians.

For example, to convert \(\frac{7\pi}{4}\) radians to degrees, you multiply \(\frac{7\pi}{4}\) by \(\frac{180}{\pi}\), resulting in 315 degrees. This conversion helps visualize the angle's position in the circle more easily.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin (0,0) on the coordinate plane. It is a fundamental tool in trigonometry because it provides a geometrical framework to understand trigonometric functions.

Each point on the unit circle corresponds to a specific angle, measured from the positive x-axis. The coordinates of these points are

• The x-coordinate: Gives \(\cos(\theta)\).
• The y-coordinate: Gives \(\sin(\theta)\).

For the angle \(\frac{7\pi}{4}\) or 315 degrees, the corresponding point on the unit circle is \( \left( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) \), highlighting that cosine provides the horizontal component (\(\frac{\sqrt{2}}{2}\)) and is positive in this quadrant.

Reference Angle

A reference angle is the acute angle that a given angle makes with the x-axis. It is always measured as a positive angle and serves to simplify trigonometric calculations.

Understanding reference angles helps in determining the function values for angles in different quadrants by relating them to familiar acute angle values. For any angle \(\theta\), the reference angle can be found based on the quadrant it lies in:

• **Quadrant I**: The reference angle is the angle itself.
• **Quadrant II**: Subtract the angle from 180 degrees.
• **Quadrant III**: Subtract 180 degrees from the angle.
• **Quadrant IV**: Subtract the angle from 360 degrees.

In the case of 315 degrees, since it lies in the fourth quadrant, the reference angle is \(360^{\circ} - 315^{\circ} = 45^{\circ}\). This reference angle helps determine that the cosine of 315 degrees is the same as the cosine of 45 degrees, but adjusted to the cosine's sign in that quadrant.