Problem 48 - Precalculus

Question

Evaluate all six trigonometric functions at $$t=-\frac{2 \pi}{3}$$

Step-by-Step Solution

Answer

\(\sin=-\frac{\sqrt{3}}{2}, \cos=-\frac{1}{2}, \tan=\sqrt{3}, \csc=-\frac{2\sqrt{3}}{3}, \sec=-2, \cot=\frac{\sqrt{3}}{3}\).

Step 1: Standard Angle Conversion

First, convert the angle \( t = -\frac{2\pi}{3} \) to a positive angle by adding \( 2\pi \). This gives us \( -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3} \). This means the angle \( \frac{4\pi}{3} \) is coterminal with \( -\frac{2\pi}{3} \).

Step 2: Identify the Reference Angle

The reference angle for \( \frac{4\pi}{3} \) can be found by subtracting \( \pi \) from the angle because \( \frac{4\pi}{3} \) is in the third quadrant. This gives us a reference angle of \( \frac{4\pi}{3} - \pi = \frac{\pi}{3} \).

Step 3: Determine the Quadrant

Since \( \frac{4\pi}{3} \) is between \( \pi \) and \( \frac{3\pi}{2} \), it is in the third quadrant. In this quadrant, sine and cosine are negative, while tangent is positive.

Step 4: Evaluate Sine and Cosine

Using the reference angle \( \frac{\pi}{3} \), the sine and cosine values are \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \) and \( \cos(\frac{\pi}{3}) = \frac{1}{2} \). Since the angle is in the third quadrant, \( \sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2} \) and \( \cos(\frac{4\pi}{3}) = -\frac{1}{2} \).

Step 5: Evaluate Tangent

Tangent is sine divided by cosine. Thus, \( \tan(\frac{4\pi}{3}) = \frac{\sin(\frac{4\pi}{3})}{\cos(\frac{4\pi}{3})} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} \).

Step 6: Evaluate Cosecant, Secant, and Cotangent

Cosecant is the reciprocal of sine, so \( \csc(\frac{4\pi}{3}) = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \). Secant is the reciprocal of cosine, so \( \sec(\frac{4\pi}{3}) = -2 \). Cotangent is the reciprocal of tangent, so \( \cot(\frac{4\pi}{3}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Key Concepts

Reference Angle

A reference angle is crucial when evaluating trigonometric functions as it helps simplify the process. It is the acute angle that a given angle makes with the x-axis. This is always a positive angle and is less than or equal to \( \frac{\pi}{2} \).

For instance, to find the reference angle of \( \frac{4\pi}{3} \), we need to consider the quadrant it resides in. Since it's in the third quadrant, the reference angle is obtained by subtracting \( \pi \) from it, resulting in \( \frac{\pi}{3} \). Understanding the reference angle allows us to determine the trigonometric function values more easily, using known table values from the first quadrant.

Quadrants in Trigonometry

The trigonometric circle is divided into four quadrants, each corresponding to different characteristics of trigonometric functions.

- **First Quadrant (0 to \( \frac{\pi}{2} \))**: All functions are positive.- **Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \))**: Only sine and cosecant are positive.- **Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \))**: Only tangent and cotangent are positive.- **Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \))**: Only cosine and secant are positive.

Recognizing the appropriate quadrant is critical as it helps determine the sign of each trigonometric function. In our problem, \( \frac{4\pi}{3} \) lies in the third quadrant, where sine and cosine are negative, while tangent is positive. This affects the evaluation of each trigonometric function at this angle.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived by taking the reciprocals of the basic trigonometric functions. These include:

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

Reciprocal functions serve as additional tools to evaluate trigonometric equations or expressions, especially in complex problems. For \( \frac{4\pi}{3} \), since sine and cosine were determined as \( -\frac{\sqrt{3}}{2} \) and \( -\frac{1}{2} \) respectively, their reciprocals yield: \( \csc(\frac{4\pi}{3}) = -\frac{2\sqrt{3}}{3} \) and \( \sec(\frac{4\pi}{3}) = -2 \). Similarly, for tangent \( \sqrt{3} \), the cotangent is \( \frac{\sqrt{3}}{3} \).

Coterminal Angles

Coterminal angles share the same terminal side when plotted in the coordinate plane. To find a coterminal angle, you can add or subtract full circles (\( 2\pi \)) to the original angle.

For \( t = -\frac{2\pi}{3} \), adding \( 2\pi \) results in \( \frac{4\pi}{3} \), making them coterminal. Being familiar with coterminal angles is helpful to transform negative angles into their positive counterparts or convert very large angles into a manageable size. This is essential because it can simplify calculations and unveil the angle's actual position in the unit circle, ensuring you solve problems more efficiently.