Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 150^{\circ} & & -\frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{3} & & & 2\\\ \hline \end{array}$$

Step-by-Step Solution

Answer

\(\sin 150^{\circ} = \frac{1}{2}\), \(\tan 150^{\circ} = -\frac{\sqrt{3}}{3}\), \(\cot 150^{\circ} = -\sqrt{3}\), \(\sec 150^{\circ} = -\frac{2\sqrt{3}}{3}\).

Step 1: Identify the Reference Angle

To find the missing trigonometric values for \(150^{\circ}\), start by identifying its reference angle. The reference angle for \(150^{\circ}\) is \(30^{\circ}\) since it is in the second quadrant and \(150^{\circ} = 180^{\circ} - 30^{\circ}\).

Step 2: Determine the Sign of Functions

Since \(150^{\circ}\) is in the second quadrant, \(\sin \theta\) is positive, \(\cos \theta\) is negative, \(\tan \theta\) is negative, \(\cot \theta\) is also negative, \(\sec \theta\) is negative, and \(\csc \theta\) is positive.

Step 3: Use Reference Angle for Function Values

Remember the basic trigonometric values for \(30^{\circ}\): \(\sin 30^{\circ} = \frac{1}{2}\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\). Use this to determine the values in the second quadrant.

Step 4: Calculate Missing Values

Using the basic values: - \(\sin 150^{\circ} = \sin 30^{\circ} = \frac{1}{2}\) (positive in the second quadrant).- \(\tan 150^{\circ} = \frac{\sin 150^{\circ}}{\cos 150^{\circ}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{\sqrt{3}}{3}\).- \(\cot 150^{\circ} = \frac{1}{\tan 150^{\circ}} = -\sqrt{3}\).- \(\sec 150^{\circ} = \frac{1}{\cos 150^{\circ}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\). - \(\csc 150^{\circ} = \frac{1}{\sin 150^{\circ}} = 2\) (already given).

Key Concepts

Reference Angle

When dealing with trigonometric functions for any angle, finding the reference angle is an essential first step. The reference angle is the acute angle (less than 90 degrees) that the given angle makes with the x-axis.
The reference angle helps us determine the trigonometric values of the angle by referring to the known values of the corresponding acute angle.
In the original problem, the angle of interest is 150 degrees. Since 150 degrees is in the second quadrant, the reference angle is found using the formula: - Reference angle = 180 degrees - 150 degrees. Hence, the reference angle is 30 degrees.
Remember, the reference angle allows us to use the standard trigonometric values of common angles like 30 degrees, 45 degrees, and 60 degrees.

Second Quadrant

Understanding which quadrant an angle lies in is crucial for determining the sign of its trigonometric functions. The coordinate plane is divided into four quadrants, and each angle will end in one of them.
For angles between 90 degrees and 180 degrees, they reside in the second quadrant. In this quadrant:

• Sine (\( \sin \theta \)) is positive.
• Cosine (\( \cos \theta \)) is negative.
• Tangent (\( \tan \theta \)) is negative.
• Cosecant (\( \csc \theta \)) is positive.
• Secant (\( \sec \theta \)) is negative.
• Cotangent (\( \cot \theta \)) is negative.

Knowing the sign of the trigonometric functions in each quadrant is key to accurately finding the exact values.

Exact Trigonometric Values

Exact trigonometric values give us precise numbers without relying on approximations or calculators. Common angles like 30, 45, and 60 degrees have well-known trigonometric values.
For example, the values for a 30-degree angle are:

• \( \sin 30^{\circ} = \frac{1}{2} \).
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).
• \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

These values are fundamental and are often memorized to assist in solving trigonometric problems quickly.
Given the reference angle is the key 30 degrees, we apply these exact values accordingly, adjusting them based on the quadrant for angles like 150 degrees.

Sign of Functions

The sign of trigonometric functions changes with each quadrant, based on the direction relative to the x and y axes.
In any trigonometry problem, especially those involving angles not in the first quadrant, addressing the sign of each function is crucial. In our problem with the angle 150 degrees, situated in the second quadrant, we know:

• The sine function is positive because it mirrors the y-axis, remaining above the x-axis.
• Cosine becomes negative as it reflects the horizontal distance or movement against the direction of positive x-values.
• Tangent, a ratio of sine to cosine, also turns negative since it involves a positive over a negative.
• The reciprocal functions, secant and cosecant, follow the signs of their associated primary functions, leading secant to be negative and cosecant to be positive.

Understanding these principles helps ensure the proper calculation of trigonometric values, aligning each function's sign with its quadrant.