Problem 28 - 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. $$t=-\frac{\pi}{6}$$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions for \(t=-\frac{\pi}{6}\) are: \(\sin(t) = -\frac{1}{2}\), \(\cos(t) = \frac{\sqrt{3}}{2}\), \(\tan(t) = -\frac{\sqrt{3}}{3}\), \(\csc(t) = -2\), \(\sec(t) = \frac{2\sqrt{3}}{3}\), and \(\cot(t) = -\sqrt{3}\).

Step 1: Sine of t

To find the sine function value, recall from the unit circle that \(\sin(\theta)\) refers to the y-coordinate. So, you have \(\sin(-\frac{\pi}{6}) = -\frac{1}{2}.\)

Step 2: Cosine of t

Likewise, the cosine function value \(\cos(\theta)\) corresponds to the x-coordinate on the unit circle. So, \(\cos(-\frac{\pi}{6}) = \frac{\sqrt{3}}{2}.\)

Step 3: Tangent of t

Tangent function is given by the ratio of sine to cosine. Thus, you have \(\tan(-\frac{\pi}{6}) = \frac{\sin(-\frac{\pi}{6})}{\cos(-\frac{\pi}{6})} = \frac{-1/2}{\sqrt{3}/2} = -\frac{\sqrt{3}}{3}.\)

Step 4: Cosecant, secant, and cotangent of t

Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. Thus, you have \(\csc(-\frac{\pi}{6}) = \frac{1}{\sin(-\frac{\pi}{6})} = -2, \sec(-\frac{\pi}{6}) = \frac{1}{\cos(-\frac{\pi}{6})} = \frac{2}{\sqrt{3}}\) (which is usually rationalized to \(\frac{2\sqrt{3}}{3}\)), and \(\cot(-\frac{\pi}{6}) = \frac{1}{\tan(-\frac{\pi}{6})} = -\sqrt{3}.\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing all the possible angles and their corresponding points on a circle with a radius of 1. It is defined as a circle centered at the origin (0,0) of the coordinate system. On the unit circle, any point can be determined using the formula \( (x, y) = (\cos(\theta), \sin(\theta)) \) where \( x \) and \( y \) are the coordinates of the point, and \( \theta \) is the angle formed with the positive x-axis.

Understanding the unit circle is essential because it provides a visual way to grasp how angles correspond to the sine and cosine values. These values can be positive or negative depending on the quadrant in which the angle terminates. For example, an angle of \( -\frac{\pi}{6} \) indicates a rotation in the negative direction from the x-axis and helps locate the correct sine and cosine values, as detailed in the steps of the solution.

Sine and Cosine

Sine and cosine are the most fundamental trigonometric functions, representing the projection of an angle's point on the y-axis and x-axis correspondingly. Specifically, the sine function \( \sin(\theta) \) gives the y-coordinate of the point on the unit circle, while the cosine function \( \cos(\theta) \) represents the x-coordinate.

In the given example, the negative angle \( -\frac{\pi}{6} \) produces a sine value of \( -\frac{1}{2} \) and a cosine value of \( \frac{\sqrt{3}}{2} \) indicating the position of the point below the x-axis for sine and to the right of the y-axis for cosine. These functions are periodic and play a critical role in oscillatory phenomena such as sound waves and tides.

Tangent and Cotangent

Tangent and cotangent, often abbreviated as \( \tan \) and \( \cot \) respectively, are functions that relate to sine and cosine. The tangent of an angle is the ratio of sine to cosine, expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Conversely, cotangent is the reciprocal of the tangent function, \( \cot(\theta) = \frac{1}{\tan(\theta)} \). These functions are particularly useful when dealing with right-angled triangles and modeling situations where angles are involved.

For the angle \( -\frac{\pi}{6} \) in the given problem, the tangent function yields \( -\frac{\sqrt{3}}{3} \), and its reciprocal is the cotangent, providing a value of \( -\sqrt{3} \) which is the steepness (slope) of the line from the origin to that point on the unit circle.

Cosecant and Secant

Cosecant and secant are the reciprocal trigonometric functions of sine and cosine, respectively. The cosecant (\( \csc \) or \( \cosec \) ) is defined as \( \frac{1}{\sin(\theta)} \) and secant (\( \sec \) ) is defined as \( \frac{1}{\cos(\theta)} \). These functions are less common but are still extensively used in calculus and higher mathematics.

In our example, cosecant of the angle \( -\frac{\pi}{6} \) is \( -2 \) and secant is \( \frac{2}{\sqrt{3}} \) which is usually written as \( \frac{2\sqrt{3}}{3} \) to rationalize the denominator. These values are useful for finding the length of the line from the origin to the point on the circle vertically and horizontally.

Radians and Degrees

Radians and degrees are two units of measure for angles. A radian is the standard unit of angular measure in mathematics, defined as the angle created by taking the radius of a circle and wrapping it along the circle's edge. One full circle is \( 2\pi \) radians, which is equivalent to 360 degrees. To convert degrees to radians, one can use the conversion ratio \( \frac{\pi}{180} \) and vice versa.

The angle \( -\frac{\pi}{6} \) provided in the problem is in radians. It's equivalent to \( -30^\circ \) in degrees. Understanding this conversion is crucial when interpreting angles and applying trigonometric functions in different contexts, such as when using a calculator that requires a specific mode for angle units.