Problem 27 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=-\frac{\pi}{3}$$

Step-by-Step Solution

Answer

The exact values are \(\sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}\), \(\cos(-\frac{\pi}{3}) = \frac{1}{2}\), \(\tan(-\frac{\pi}{3}) = -\sqrt{3}\), \(\csc(-\frac{\pi}{3}) = -\frac{2}{\sqrt{3}}\), \(\sec(-\frac{\pi}{3}) = 2\), and \(\cot(-\frac{\pi}{3}) = -\frac{1}{\sqrt{3}}\).

Step 1: Determine the sine function value

In the unit circle, the sine of any angle \(t\) corresponds to the y-coordinate. For \(t = -\frac{\pi}{3}\), the sine function value is negative from the definition of sine in the fourth quadrant. Therefore, we find \(\sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}\).

Step 2: Determine the cosine function value

In unit circle, the cosine of any angle \(t\) corresponds to the x-coordinate. For \(t = -\frac{\pi}{3}\), the cosine function value is positive from the definition of cosine in the fourth quadrant. Therefore, we find \(\cos(-\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}\).

Step 3: Determine the tangent function value

The tangent of any angle \(t\) is defined by the ratio of the sine to the cosine. Therefore, \(\tan(-\frac{\pi}{3}) = \frac{\sin(-\frac{\pi}{3})}{\cos(-\frac{\pi}{3})} = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3}\). In this case, the tangent function is negative as the sine and cosine have different signs.

Step 4: Determine the values of the other trigonometric functions

The cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and tangent respectively. Therefore, \(\csc(-\frac{\pi}{3}) = \frac{1}{\sin(-\frac{\pi}{3})} = -\frac{2}{\sqrt{3}}\), \(\sec(-\frac{\pi}{3}) = \frac{1}{\cos(-\frac{\pi}{3})} = 2\), and \(\cot(-\frac{\pi}{3}) = \frac{1}{\tan(-\frac{\pi}{3})} = -\frac{1}{\sqrt{3}}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, defined as a circle with a radius of one unit and its center at the origin of a coordinate system. It's a powerful tool for understanding trigonometric functions because each point on the unit circle corresponds to an angle measured from the positive x-axis, and its coordinates define the values of sine and cosine for that angle.

For example, when working with the angle \(t = -\frac{\pi}{3}\), its location on the unit circle can help us find the exact trigonometric values. By drawing the angle in standard position, we see that the terminal side falls in the fourth quadrant, where the sine values are negative and the cosine values are positive. This is why \(\sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}\) and \(\cos(-\frac{\pi}{3}) = \frac{1}{2}\).

Trigonometric Ratios

Trigonometric ratios are the basis of trigonometry, relating the angles of a triangle to the lengths of its sides. In the context of the unit circle, these ratios define the sine, cosine, and tangent functions for any given angle.

Understanding Sine and Cosine

The sine of an angle corresponds to the y-coordinate on the unit circle, while the cosine relates to the x-coordinate. For the angle \(t = -\frac{\pi}{3}\), we determined that \(\sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}\) and \(\cos(-\frac{\pi}{3}) = \frac{1}{2}\).

Defining Tangent

The tangent function is the ratio of the sine to the cosine. Given \(\tan(-\frac{\pi}{3}) = \frac{\sin(-\frac{\pi}{3})}{\cos(-\frac{\pi}{3})} = -\sqrt{3}\), we use our previous findings to calculate this value, illustrating the relationship between these trigonometric ratios.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide alternative ways of expressing the relationships between the sides of a right-angle triangle and are essential in finding the values of trigonometric functions that aren't directly available from the unit circle.

The cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)) are the reciprocals of the sine, cosine, and tangent functions, respectively. For \(t = -\frac{\pi}{3}\), we have \(\csc(-\frac{\pi}{3}) = \frac{1}{\sin(-\frac{\pi}{3})} = -\frac{2}{\sqrt{3}}\), \(\sec(-\frac{\pi}{3}) = \frac{1}{\cos(-\frac{\pi}{3})} = 2\), and \(\cot(-\frac{\pi}{3}) = \frac{1}{\tan(-\frac{\pi}{3})} = -\frac{1}{\sqrt{3}}\), demonstrating how these functions are calculated and their importance in trigonometry.