Problem 85 - A Graphical Approach to Precalculus with Limits

Question

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{1}{2}, \cos s=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

\( \tan s = \frac{\sqrt{3}}{3} \), \( \cot s = \sqrt{3} \), \( \sec s = \frac{2\sqrt{3}}{3} \), \( \csc s = 2 \).

Step 1: Understanding Given Values

We are given \( \sin s = \frac{1}{2} \) and \( \cos s = \frac{\sqrt{3}}{2} \). From these values, we can identify that angle \( s \) corresponds to a common angle used in trigonometry.

Step 2: Identify Common Angle

The values given, \( \sin s = \frac{1}{2} \) and \( \cos s = \frac{\sqrt{3}}{2} \), correspond to the angle \( s = 30^{\circ} \) or \( s = \frac{\pi}{6} \) radians, which is a common angle in the unit circle.

Step 3: Calculate \( \tan s \)

The tangent of an angle \( s \) is calculated by the ratio of sine to cosine: \( \tan s = \frac{\sin s}{\cos s} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \). It is often simplified to \( \frac{\sqrt{3}}{3} \) by rationalizing the denominator.

Step 4: Calculate \( \cot s \)

The cotangent is the reciprocal of tangent: \( \cot s = \frac{1}{\tan s} = \sqrt{3} \).

Step 5: Calculate \( \sec s \)

The secant of an angle is the reciprocal of cosine: \( \sec s = \frac{1}{\cos s} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \). When rationalized, this becomes \( \frac{2\sqrt{3}}{3} \).

Step 6: Calculate \( \csc s \)

The cosecant is the reciprocal of sine: \( \csc s = \frac{1}{\sin s} = \frac{1}{\frac{1}{2}} = 2 \).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry that helps us understand angles and their corresponding values for sine, cosine, and tangent. It is called the "unit circle" because it has a radius of one unit. This allows for easy computations of trigonometric functions using the coordinates of points on the circle.

Here’s how it works:

• The center of the unit circle is at the origin (0,0) on the coordinate plane.
• Angles are measured from the positive x-axis.
• The circle's circumference contains points with coordinates (\(x, y\)), where \(x = \cos \theta\) and \(y = \sin \theta\).

These coordinates enable you to find trigonometric values directly. For example, the coordinates for \(30^{\circ}\) or \(\frac{\pi}{6}\) are (\(\frac{\sqrt{3}}{2}, \frac{1}{2}\)), which means \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin \frac{\pi}{6} = \frac{1}{2}\). Understanding this concept allows you to quickly identify common angles and their trigonometric values.

Common Angles

In trigonometry, certain angles are frequently used because they have simple and well-known trigonometric ratios. These are known as common angles and they include \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\).

• \(30^{\circ}\)has sine, cosine values: \(\sin 30^{\circ} = \frac{1}{2}, \, \cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
• Being familiar with these ratios allows quick identification without calculations.

When given \(\sin s = \frac{1}{2}, \, \cos s = \frac{\sqrt{3}}{2}\), recognizing these values can help you conclude that the angle \(s\) is \(30^{\circ}\) or \(\frac{\pi}{6}\). This is because these sine and cosine values exactly match those of the \(30^{\circ}\) angle. Common angles are like shortcuts in trigonometry, allowing easy navigation through exercises without a calculator.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are based on the main trigonometric ratios but involve flipping their values. Understanding these functions helps solve more complex trigonometric problems.

Here's a breakdown:

• \(\csc\) (cosecant) is the reciprocal of \(\sin\): \(\csc s = \frac{1}{\sin s}\).
• \(\sec\) (secant) is the reciprocal of \(\cos\): \(\sec s = \frac{1}{\cos s}\).
• \(\cot\) (cotangent) is the reciprocal of \(\tan\): \(\cot s = \frac{1}{\tan s}\).

For instance, if \(\sin s = \frac{1}{2}\), then \(\csc s = 2\). Similarly, if \(\cos s = \frac{\sqrt{3}}{2}\), the secant becomes \(\sec s = \frac{2\sqrt{3}}{3}\) after rationalization.

Reciprocal functions provide alternate ways to view an angle's trigonometric ratios and are invaluable in both simplifying and solving trigonometric equations. Mastering them adds tremendous flexibility to your trigonometric toolbox.