Problem 7 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \cos \frac{3 \pi}{4}} & {\text { (b) } \cos \frac{5 \pi}{4}} & {\text { (c) } \cos \frac{7 \pi}{4}}\end{array} $$

Step-by-Step Solution

Answer

\( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), \( \cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \), \( \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 1: Identify Angle in Standard Position

For the angle \( \frac{3\pi}{4} \), notice that it lies in the second quadrant. In standard position, it's important to find the reference angle, which can be calculated as \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).

Step 2: Use Trigonometric Identity to Find Cosine

In the second quadrant, cosine values are negative, therefore \( \cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \).

Step 3: Repeat for Second Angle

For \( \frac{5\pi}{4} \), which is in the third quadrant, the reference angle is \( \frac{5\pi}{4} - \pi = \frac{\pi}{4} \). Here, cosine is also negative, hence \( \cos \frac{5\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \).

Step 4: Repeat for Third Angle

For \( \frac{7\pi}{4} \), this angle is in the fourth quadrant and the reference angle is \( 2\pi - \frac{7\pi}{4} = \frac{\pi}{4} \). In the fourth quadrant, cosine is positive, hence \( \cos \frac{7\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Cosine

In trigonometry, the cosine of an angle is a fundamental ratio that relates the lengths of sides in a right triangle. Specifically, for a given angle in a right triangle, cosine is the ratio of the length of the adjacent side to the hypotenuse. Here is how it is usually expressed:

• Adjacent side: The side that forms the angle along with the hypotenuse.
• Hypotenuse: The longest side of the triangle, opposite the right angle.

This can be written mathematically as:\[\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\]In a unit circle, cosine corresponds to the x-coordinate of the point on the circle at a given angle. This means the cosine value ranges between -1 and 1.
Cosine is often used to find missing side lengths or calculate angles in trigonometry, always yielding crucial insights into the angle's properties and placements on a unit circle.

Reference Angle

The reference angle is a critical concept in determining trigonometric values for angles outside the first quadrant. It is the smallest angle that the terminal side of an angle makes with the x-axis. The idea is that any angle's sine, cosine, and tangent values depend on its position relative to 90°, 180°, 270°, and 360° (or their radian equivalents).### Calculating the Reference AngleDetermining a reference angle depends on the quadrant of the given angle. Here are some tips:

• First Quadrant: The reference angle is the angle itself as it's between 0 to \(\frac{\pi}{2}\) .
• Second Quadrant: Subtract the angle from \(\pi\) (180°) to find the reference angle. For example, \(\pi - \theta\).
• Third Quadrant: Subtract \(\pi\) from the angle to find it. Use \(\theta - \pi\).
• Fourth Quadrant: Subtract the angle from \(2\pi\) (360°), as in \(2\pi - \theta\).

Understanding reference angles is useful because it enables the application of known sine and cosine values to any angle, preserving the sign appropriate for the quadrant.

Quadrant Analysis

Quadrant analysis in trigonometry revolves around determining the sign and value of trigonometric functions based on the location of an angle within the four quadrants of the coordinate plane. Each quadrant has specific characteristics:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, while cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, and cosine is positive.

### Applying Quadrant AnalysisWhen determining the cosine value for angles like \(\frac{3\pi}{4}\), \(\frac{5\pi}{4}\), and \(\frac{7\pi}{4}\), it is crucial to know their respective quadrants:

• \(\frac{3\pi}{4}\): In the second quadrant, making cosine negative.
• \(\frac{5\pi}{4}\): In the third quadrant where cosine is negative.
• \(\frac{7\pi}{4}\): In the fourth quadrant where cosine is positive.

Understanding each quadrant's properties allows you to deduce not just the correct sign for trigonometric functions, but also a symmetry that helps in calculating exact values.