Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 30^{\circ} & \frac{1}{2} & \frac{\sqrt{3}}{2} & & & \frac{2 \sqrt{3}}{3} & 2 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\(\tan(30^\circ) = \frac{\sqrt{3}}{3}\), \(\cot(30^\circ) = \sqrt{3}\)

Step 1: Recap Known Values

We know that \( \sin(30^\circ) = \frac{1}{2} \) and \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \). These are standard trigonometric values based on the unit circle.

Step 2: Calculate \( \tan(30^\circ) \)

The tangent of an angle \( \theta \) in a right triangle is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). For \( 30^\circ \), \( \tan(30^\circ) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Step 3: Calculate \( \cot(30^\circ) \)

The cotangent is the reciprocal of the tangent. Thus, \( \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{3}} = \sqrt{3} \).

Step 4: Verify Provided Value for \( \sec(30^\circ) \)

The secant function is the reciprocal of the cosine function, \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \sec(30^\circ) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2 \sqrt{3}}{3} \), which matches the provided table value.

Step 5: Verify Provided Value for \( \csc(30^\circ) \)

The cosecant function is the reciprocal of the sine function, \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, \( \csc(30^\circ) = \frac{1}{\frac{1}{2}} = 2 \), which matches the provided table value.

Key Concepts

Unit Circle

The unit circle is a key concept in trigonometry, offering a visual and practical method to find exact values for trigonometric functions. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate system. When we discuss angles in trigonometry, they are often measured in standard position, where the initial side lies along the positive x-axis.
The coordinates of points on the unit circle give us the exact values for the sine and cosine of any angle.

• The x-coordinate represents \( \cos(\theta) \)
• The y-coordinate represents \( \sin(\theta) \)

For example, at an angle of 30 degrees, the coordinates would be \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \). These coordinates provide the exact values for \( \sin \theta \) and \( \cos \theta \), which are crucial for calculating other trigonometric functions.
This intuitive method simplifies finding and remembering trigonometric values without the need for a calculator.

Exact Values

Exact values in trigonometry are values derived from well-known angles that make calculating trigonometric functions straightforward and precise. Common angles include 30°, 45°, and 60°.
These angles appear frequently in geometry, especially in right triangles where they create predictable trigonometric ratios.
Here are some exact value highlights:

• For \( 30^{\circ} \): \( \sin(30^{\circ}) = \frac{1}{2} \, \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \, \tan(30^{\circ}) = \frac{\sqrt{3}}{3} \)
• For \( 45^{\circ} \): \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \, \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \, \tan(45^{\circ}) = 1 \)
• For \( 60^{\circ} \): \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \, \cos(60^{\circ}) = \frac{1}{2} \, \tan(60^{\circ}) = \sqrt{3} \)

Knowing these exact values helps students fill in trigonometric tables easily and accurately.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions but as their name suggests, are the reciprocals. These include the secant, cosecant, and cotangent functions. Understanding them can deepen your grasp of trigonometric relationships.

• Secant (\( \sec \theta \)) is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Cosecant (\( \csc \theta \)) is the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Cotangent (\( \cot \theta \)) is the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

For instance, when dealing with \(30^{\circ}\), using reciprocal identities:

• \( \sec(30^{\circ}) = \frac{2 \sqrt{3}}{3} \)
• \( \csc(30^{\circ}) = 2 \)
• \( \cot(30^{\circ}) = \sqrt{3} \)

Mastering these concepts can help solve more complex trigonometric equations and problems.