Problem 70 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \cot \frac{7 \pi}{4} $$

Step-by-Step Solution

Answer

The exact value of \(\cot (\frac{7 \pi}{4})\) is 1.

Step 1: Determine the reference angle

The referent angle for \(\frac{7 \pi}{4}\) is obtained by evaluating the modulus of the angle and subtracting any full rotations. A full rotation in radians is equal to \(2\pi\). Hence, \(\text{Reference Angle} = \frac{7 \pi}{4} \mod 2\pi = \frac{\pi}{4}\)

Step 2: Evaluate the cotangent

The cotangent is the reciprocal of the tangent function. The tangent of our reference angle, \(\frac{\pi}{4}\), is 1, therefore, \(\cot (\frac{\pi}{4}) = \frac{1}{\tan(\frac{\pi}{4})} = 1\)

Step 3: Conclusion

The exact value of \(\cot (\frac{7 \pi}{4})\) is 1.

Key Concepts

Cotangent

In trigonometry, the cotangent is an important function that represents the reciprocal of the tangent. When you're working with the tangent of an angle, you're dealing with the ratio of the opposite side to the adjacent side in a right triangle. Conversely, the cotangent gives us the ratio of the adjacent side to the opposite side.

This flips the perspective: instead of looking at how tall something is relative to how far we are from its base (tangent), we look at how far we are from the base relative to its height (cotangent). For any angle \theta, the cotangent is defined as \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

To find the cotangent of an angle in a right triangle, you can either take the reciprocal of the tangent of that angle or use the sides of the triangle directly if they’re known. For the angle \( \frac{7\pi}{4} \) the cotangent can be found using its reference angle, leading us to the exact value without the need for a calculator.

Exact Trigonometric Values

In trigonometry, exact values are specific numbers, like \(\sqrt{2}\), \(1\), or \(\frac{1}{2}\), that represent the trigonometric ratios for angles that are commonly found in uniquely constructed triangles, such as the 45°-45°-90° triangle or the 30°-60°-90° triangle.

These exact values are crucial because they allow us to solve trigonometric problems without approximation, ensuring precision in fields like engineering and physics. The exercise in question asks for the exact value of the cotangent function at \(\frac{7\pi}{4}\) radians. Because \(\frac{7\pi}{4}\) is a standard angle on the unit circle, corresponding to 315°, its exact value can be derived from known trigonometric ratios for its associated reference angle, \(\frac{\pi}{4}\).

Knowing these exact values also provides a strong foundation for understanding trigonometric identities and for solving more complex trigonometric equations.

Radians

Radians represent a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians use the length of the radius to define the measure of an angle. A full circle in radians is \(2\pi\), as the circumference of a circle is \(2\pi\) times its radius.

Working with radians can make calculations involving angular motion or lengths of arcs more straightforward, because they relate directly to the radius. In trigonometry, radians are the preferred unit because they simplify many formulas and make it easier to derive exact trigonometric values and to understand the nature of trigonometric functions as the angle changes.

For example, in the given exercise, the angle \(\frac{7\pi}{4}\) is expressed in radians, which immediately tells us it's related to a full circle minus \(\frac{\pi}{4}\), making it easier to work out its reference angle and accordingly, to find the cotangent's exact value.