Problem 3 - Calculus Single Variable

Question

Calculate each of the six trigonometric functions at angle \(\theta\) without using a calculator. \(\theta=2 \pi / 3\)

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: Identify the Reference Angle

Recognize that the angle \( \theta = \frac{2\pi}{3} \) is in the second quadrant. The reference angle is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).

Step 2: Determine the Sine

The sine function is positive in the second quadrant. Since the reference angle is \( \frac{\pi}{3} \), \( \sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).

Step 3: Calculate the Cosine

Cosine is negative in the second quadrant. The cosine of the reference angle is \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \). Therefore, \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \).

Step 4: Find the Tangent

Tangent is sine over cosine. Thus, \( \tan\left(\frac{2\pi}{3}\right) = \frac{\sin\left(\frac{2\pi}{3}\right)}{\cos\left(\frac{2\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \).

Step 5: Calculate Cosecant

Cosecant is the reciprocal of sine. Therefore, \( \csc\left(\frac{2\pi}{3}\right) = \frac{1}{\sin\left(\frac{2\pi}{3}\right)} = \frac{2}{\sqrt{3}} \) which simplifies to \( \frac{2\sqrt{3}}{3} \).

Step 6: Determine Secant

Secant is the reciprocal of cosine. Thus, \( \sec\left(\frac{2\pi}{3}\right) = \frac{1}{\cos\left(\frac{2\pi}{3}\right)} = -2 \).

Step 7: Find Cotangent

Cotangent is the reciprocal of tangent. Thus, \( \cot\left(\frac{2\pi}{3}\right) = \frac{1}{\tan\left(\frac{2\pi}{3}\right)} = -\frac{1}{\sqrt{3}} \), which simplifies to \( -\frac{\sqrt{3}}{3} \).

Key Concepts

Reference Angle

In trigonometry, the reference angle is a key concept that helps in finding trigonometric function values for any angle. A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis. For the angle \( \theta = \frac{2\pi}{3} \), the reference angle is determined by subtracting this angle from \( \pi \). Therefore, \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \). This means the reference angle is the equivalent angle in the first quadrant that shares the same trigonometric properties but may differ in sign due to the quadrant placement.
Understanding the reference angle is crucial because it lets you use familiar first-quadrant values to find values in different quadrants, adjusting for sign based on the quadrant.

Quadrants

The coordinate plane is divided into four quadrants, which help us determine the sign of trigonometric functions. The angle \( \theta = \frac{2\pi}{3} \) is located in the second quadrant. Each quadrant determines whether the sine, cosine, and tangent functions are positive or negative:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive; cosine and tangent are negative.
• Third Quadrant: Tangent is positive; sine and cosine are negative.
• Fourth Quadrant: Cosine is positive; sine and tangent are negative.

For \( \frac{2\pi}{3} \), we observe it in the second quadrant, where sine remains positive, and cosine and tangent are negative. Recognizing the quadrant allows you to properly assign signs to each trigonometric value, ensuring accuracy in calculations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions play an important role in expanding our understanding of trigonometry. They provide another perspective of the basic sine, cosine, and tangent:

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

For \( \theta = \frac{2\pi}{3} \), once you know the sine, cosine, and tangent, finding the reciprocal functions is straightforward. For instance, \( \csc(\frac{2\pi}{3}) = \frac{2\sqrt{3}}{3} \), \( \sec(\frac{2\pi}{3}) = -2 \), and \( \cot(\frac{2\pi}{3}) = -\frac{\sqrt{3}}{3} \). These functions are often needed in various mathematical applications and calculations.

Tangent as Sine over Cosine

The tangent of an angle is an interesting concept because it derives from the sine and cosine of the angle, specifically as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship showcases how tangent can be calculated using the two primary functions, providing a deeper understanding of its behavior in different quadrants.
For \( \theta = \frac{2\pi}{3} \), knowing that \( \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \) and \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \), we calculate \( \tan(\frac{2\pi}{3}) = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \).
This shows how tangent turns negative in the second quadrant due to the division of positive sine by negative cosine. Understanding this connection is vital for solving trigonometric equations and understanding angle behaviors.