Problem 89 - 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. $$\sin s=-\frac{\sqrt{3}}{2}, \cos s=\frac{1}{2}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Given Values

We are given that \( \sin s = -\frac{\sqrt{3}}{2} \) and \( \cos s = \frac{1}{2} \). These values correspond to a point on the unit circle, which typically relates to a right triangle. Considering the given sine is negative and cosine is positive, \( s \) is situated in the fourth quadrant.

Step 2: Calculating Tan s

The tangent of angle \( s \) is defined as \( \tan s = \frac{\sin s}{\cos s} \). Substituting the given values, we have:\[ \tan s = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3} \].

Step 3: Calculating Cot s

The cotangent is the reciprocal of the tangent, \( \cot s = \frac{1}{\tan s} \). From the previous step, \( \tan s = -\sqrt{3} \), therefore:\[ \cot s = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3} \].

Step 4: Calculating Sec s

The secant of \( s \) is the reciprocal of cosine, \( \sec s = \frac{1}{\cos s} \). Substituting the given cosine value, we have:\[ \sec s = \frac{1}{\frac{1}{2}} = 2 \].

Step 5: Calculating Csc s

The cosecant is the reciprocal of sine, \( \csc s = \frac{1}{\sin s} \). Given \( \sin s = -\frac{\sqrt{3}}{2} \), we find:\[ \csc s = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \].

Key Concepts

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle allows us to define trigonometric functions for all real numbers. When a point

• has coordinates (\(\cos \theta\), \( \sin \theta\)),
• \(\theta\) represents an angle measured in radians from the positive x-axis.

The unit circle helps us understand the behavior of sine and cosine functions, especially how they repeat every

• \(2\pi\) radians.
• This periodicity makes it easy to identify quadrant signs for sine and cosine: both positive in the first quadrant, cosine positive and sine negative in the fourth quadrant, etc.

The unit circle also connects with right triangles. In this circle, for any angle \(\theta\), the horizontal leg of a corresponding right triangle is given by \(\cos \theta\). Likewise, its vertical leg is \( \sin \theta \).

Trigonometric Functions

Trigonometric functions are functions of an angle and are key to understanding relationships in geometry. These include:

• Sine (\(\sin\)),
• Cosine (\(\cos\)),
• Tangent (\(\tan\)),
• Cosecant (\(\csc\)),
• Secant (\(\sec\)),
• Cotangent (\(\cot\)).

Each function is a ratio of different sides in a right triangle.

• For example, \(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\),
• and is also the length of the opposite side over the adjacent side in a right triangle.
• \(\sec \theta\) and \(\csc \theta\) are reciprocals of \(\cos \theta\) and \(\sin \theta\) respectively.

These relationships are critical, as they reveal how angles and side lengths interact in triangles as well as on the unit circle. Using the reciprocal identities can simplify solving problems and developing insights into trigonometric equations.

Right Triangle

The right triangle is a fundamental tool in trigonometry, providing a visual method to comprehend the trigonometric ratios. Comprised of one right angle (90°) and two other angles, the relationships allow for

• trigonometric functions to relate one angle's measure to the proportions of the sides.

We define the sides relative to a given angle:

• The side opposite the angle,
• The side adjacent to the angle,
• The hypotenuse, which is opposite the right angle and the longest side.

In trigonometry, knowing any two sides allows calculation of the third side using functions like \(\sin\), \(\cos\), and \(\tan\). This foundational structure reveals essential insights,

• such as how the unit circle originates from these triangles,
• since the circle's radius acts like the triangle's hypotenuse, simplified to 1 unit length.