Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 135^{\circ} & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & & & -\sqrt{2} & \sqrt{2} \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\( \tan 135° = -1 \) and \( \cot 135° = -1 \).

Step 1: Understanding the Angle

135° is located in the second quadrant of the unit circle, where the sine value is positive and the cosine value is negative. This helps explain why \( \sin 135° = \frac{\sqrt{2}}{2} \) and \( \cos 135° = -\frac{\sqrt{2}}{2} \).

Step 2: Finding \( \tan \theta \) for 135°

The tangent of an angle is given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substituting the values we have, \( \tan 135° = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1 \).

Step 3: Calculating \( \cot \theta \) for 135°

The cotangent is the reciprocal of the tangent: \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \tan 135° = -1 \), \( \cot 135° = -1 \).

Step 4: Using Given Values for Other Functions

For \( \sec \theta \) and \( \csc \theta \), we already have the values: \( \sec 135° = -\sqrt{2} \) and \( \csc 135° = \sqrt{2} \) based on the table. They are reciprocals of \( \cos 135° \) and \( \sin 135° \), respectively.

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry that helps us understand how trigonometric functions like sine and cosine work. It is a circle with a radius of 1 centered at the origin of a coordinate plane. Every point on the unit circle corresponds with

• An angle measured in degrees or radians
• Coordinates \((x, y)\)

These coordinates directly relate to the values of trigonometric functions for that angle. When an angle is plotted on the unit circle, the x-coordinate gives \( ext{cosine}\) and the y-coordinate gives \( ext{sine}\). For example, for an angle of 135°, the \((x, y)\) coordinates are \(-\frac{\sqrt{2}}{2}\) and \(\frac{\sqrt{2}}{2}\) respectively. This explains why the \( ext{sine}\) of 135° is \(\frac{\sqrt{2}}{2}\) and the \( ext{cosine}\) is \(-\frac{\sqrt{2}}{2}\).

Angle Quadrants

The unit circle is divided into four quadrants, important for determining the sign of trigonometric functions.

• The first quadrant includes angles from 0° to 90°.
• The second quadrant includes angles from 90° to 180°.
• The third quadrant encompasses angles from 180° to 270°.
• The fourth quadrant ranges from 270° to 360°.

The location of 135° is in the second quadrant, where

• \( ext{sine values are positive}\)
• \( ext{cosine and tangent values are negative}\).

This quadrant's specifics explain the positive sine and negative cosine for 135°.

Tangent and Cotangent

Tangent and cotangent are fundamental trigonometric functions that are closely related to sine and cosine.

Tangent Function

The tangent of an angle \(\theta\) is calculated using the formula \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This means it is a ratio of sine to cosine. For 135°, we compute \(\tan 135° = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1\).

Cotangent Function

Cotangent, on the other hand, is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta}\). Given that \(\tan 135° = -1\), the cotangent is \(\cot 135° = -1\). This reciprocal relationship is key to understanding these two functions together.

Sine and Cosine

Sine and cosine are the basic building blocks of trigonometry, representing the coordinates on the unit circle. The sine of an angle \(\theta\) is the y-coordinate, while the cosine is the x-coordinate. These functions relate to any angle on the unit circle as follows:

• Sine and cosine values range between \(-1\) and \(1\).
• They depict wave-like behaviors, making them useful in modeling cyclical phenomena.

For 135°,

• The sine value is \(\frac{\sqrt{2}}{2}\)
• The cosine value is \(-\frac{\sqrt{2}}{2}\).

These values illustrate the symmetry and periodicity of trigonometric functions.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary functions, sine, cosine, and tangent. They include secant, cosecant, and cotangent.

• Secant ( \(\sec\theta\) ) is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
• Cosecant ( \(\csc\theta\) ) is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\).
• We already highlighted that cotangent is the reciprocal of tangent.

For 135°,

• The secant value is \(-\sqrt{2}\) (since its reciprocal involves the negative cosine value).
• Cosecant is \(\sqrt{2}\) because it reciprocates the positive sine value.

Understanding these functions helps expand the versatility of trigonometry in solving more complex problems.