Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 120^{\circ} & \frac{\sqrt{3}}{2} & & -\sqrt{3} & & & \frac{2 \sqrt{3}}{3} \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\( \cos 120^{\circ} = -\frac{1}{2}, \sec 120^{\circ} = -2, \cot 120^{\circ} = -\frac{\sqrt{3}}{3} \).

Step 1: Identify Known Values

The given table provides the value of \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \) and \( \tan 120^{\circ} = -\sqrt{3} \). It also provides \( \csc 120^{\circ} = \frac{2 \sqrt{3}}{3} \). We'll use these values to find the remaining trigonometric functions.

Step 2: Use Trigonometric Identities

Recall the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Using \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \), we find \( \cos 120^{\circ} \) by substituting into the identity:

Step 3: Calculate \( \cos 120^{\circ} \)

From \( \sin^2 120^{\circ} + \cos^2 120^{\circ} = 1 \), substitute \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \) to get:\[ \left( \frac{\sqrt{3}}{2} \right)^2 + \cos^2 120^{\circ} = 1 \]Simplifying, we have:\[ \frac{3}{4} + \cos^2 120^{\circ} = 1 \]\[ \cos^2 120^{\circ} = \frac{1}{4} \]Since \( 120^{\circ} \) is in the second quadrant where cosine is negative, \( \cos 120^{\circ} = -\frac{1}{2} \).

Step 4: Calculate \( \sec 120^{\circ} \)

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). With \( \cos 120^{\circ} = -\frac{1}{2} \), we find:\[ \sec 120^{\circ} = \frac{1}{-\frac{1}{2}} = -2 \].

Step 5: Calculate \( \cot 120^{\circ} \)

Recall the identity \( \cot \theta = \frac{1}{\tan \theta} \). Using \( \tan 120^{\circ} = -\sqrt{3} \), we find:\[ \cot 120^{\circ} = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3} \].

Key Concepts

Sine and Cosine

Sine and cosine are two fundamental trigonometric functions. They are used to relate the angles and sides of a right triangle. In this exercise, we are dealing with the angle \(120^{\circ}\). This specific angle is located in the second quadrant.

Here’s an important property in the second quadrant: the sine of angles is positive, while the cosine is negative. This is because sine relates to the vertical component (or opposite side), while cosine relates to the horizontal component (or adjacent side) of the angle. Therefore, for \(120^{\circ}\),

• \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \cos 120^{\circ} = -\frac{1}{2} \)

It is crucial to leverage these values when solving trigonometric problems without the use of a calculator. Knowing the signs and values for these key angles allows us to solve problems more efficiently.

Tangent and Cotangent

Tangent and cotangent functions are derived from sine and cosine. While tangent is the ratio of sine over cosine, cotangent is the inverse of tangent. These functions help in identifying the slope of the angle when applied on a coordinate plane.

For the angle \(120^{\circ}\):

• The tangent, calculated as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), results in \( \tan 120^{\circ} = -\sqrt{3} \). This negative value is aligned with the quadrant's properties, where tangent is negative.
• On the other hand, cotangent is \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, \( \cot 120^{\circ} = -\frac{\sqrt{3}}{3} \), which is also negative in the second quadrant.

Understanding these relationships helps us to better comprehend all the properties of angles in different quadrants.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities are useful tools in simplifying expressions and solving for unknown values.

One of the most important identities is

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

For \(120^{\circ}\), we use this identity to find \( \cos 120^{\circ} \). Given \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \), substituting into the identity gives:

• \( \cos^2 120^{\circ} = \frac{1}{4} \), thus \(\cos 120^{\circ} = -\frac{1}{2} \).

Other identities include relationships like

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)

These formulas enable us to derive values from given ones, broadening our toolkit for tackling different trigonometric challenges.