Problem 21 - Contemporary Precalculus

Question

In Exercises \(21-25,\) evaluate all six trigonometric finctions at the given number without using a calculator. $$\frac{4 \pi}{3}$$

Step-by-Step Solution

Answer

Answer: The values of the six trigonometric functions at \(\frac{4 \pi}{3}\) are: $$\sin\left(\frac{4\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \tan\left(\frac{4\pi}{3}\right) = -\sqrt{3}$$ $$\csc\left(\frac{4\pi}{3}\right) = \frac{2\sqrt{3}}{3}, \quad \sec\left(\frac{4\pi}{3}\right) = -2, \quad \cot\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{3}$$

Step 1: Determine the quadrant of the angle

To determine the quadrant of the angle, we observe that \(0 < \frac{4 \pi}{3} < 2\pi\). The given angle \(\frac{4 \pi}{3}\) can also be written as \(\frac{4}{3}\pi\). Since \(1 < \frac{4}{3} \pi < 2\pi\), the angle lies between \(\pi\) and \(2\pi\), which means it is in the third quadrant.

Step 2: Find the reference angle

The reference angle is the angle formed between the given angle and the x-axis. In the third quadrant, the reference angle is given by: $$\text{reference angle} = \text{Given angle} - \pi$$ In our case: $$\text{reference angle} = \frac{4 \pi}{3} - \pi = \frac{4 \pi}{3} - \frac{3 \pi}{3} = \frac{\pi}{3}$$

Step 3: Evaluate the trigonometric functions

In the third quadrant, sine and tangent are positive, and cosine is negative. We can now use the reference angle, \(\frac{\pi}{3}\), to evaluate the six trigonometric functions: 1. \(\sin(\frac{4\pi}{3}) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\) 2. \(\cos(\frac{4\pi}{3}) = -\cos(\frac{\pi}{3}) = -\frac{1}{2}\) 3. \(\tan(\frac{4\pi}{3}) = \frac{\sin(\frac{4\pi}{3})}{\cos(\frac{4\pi}{3})} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}\) 4. \(\csc(\frac{4\pi}{3}) = \frac{1}{\sin(\frac{4\pi}{3})} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\) 5. \(\sec(\frac{4\pi}{3}) = \frac{1}{\cos(\frac{4\pi}{3})} = \frac{1}{-\frac{1}{2}} = -2\) 6. \(\cot(\frac{4\pi}{3}) = \frac{1}{\tan(\frac{4\pi}{3})} = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3}\) Thus, we have obtained the values for all six trigonometric functions at the given angle: $$\sin\left(\frac{4\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \tan\left(\frac{4\pi}{3}\right) = -\sqrt{3}$$ $$\csc\left(\frac{4\pi}{3}\right) = \frac{2\sqrt{3}}{3}, \quad \sec\left(\frac{4\pi}{3}\right) = -2, \quad \cot\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{3}$$

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane. This simple setup allows us to define trigonometric functions in an easy and intuitive way.

On the unit circle, any angle is represented as a point with coordinates \((\cos(\theta), \sin(\theta))\). This means that the x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. Thus, you can find the sine and cosine of any angle using its position on the unit circle.

• A full circle is \(2\pi\) radians or 360 degrees.
• The circle is divided into quadrants, which help determine the sign of trigonometric functions.

Understanding the unit circle can also help visualize the periodicity and symmetry of trigonometric functions.

Reference Angle

A reference angle is an acute angle formed by the terminal side of a given angle and the x-axis. It is a helpful tool to evaluate trigonometric functions for angles beyond the first quadrant.

To find the reference angle, you take the absolute difference between the given angle and the nearest x-axis. The idea is to convert any angle into one that is easy to work with.

• If the angle is in the second quadrant, subtract the angle from \(\pi\).
• For the third quadrant, subtract \(\pi\) from the angle.
• In the fourth quadrant, subtract the angle from \(2\pi\).

Regardless of the original angle's quadrant, the reference angle is always acute and helps use the basic angles on the unit circle to find trigonometric values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities can simplify expressions and solve trigonometric equations. Understanding them is key to advancing in mathematics.

Some essential trigonometric identities include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal Identities:
  • \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
  • \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
  • \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Co-Function Identities: These relate the trigonometric functions of complementary angles.

These identities allow for the simplification and transformation of trigonometric expressions and are particularly useful when solving trigonometric equations.

Quadrant Angles

Quadrant angles are critical when evaluating the sign of trigonometric functions. The coordinate plane is divided into four quadrants, helping us understand where an angle lies and the subsequent signs of its trigonometric values.

• First Quadrant (0 to \(\frac{\pi}{2}\)): All trigonometric functions are positive.
• Second Quadrant (\(\frac{\pi}{2}\) to \(\pi\)): Sine is positive, and cosine and tangent are negative.
• Third Quadrant (\(\pi\) to \(\frac{3\pi}{2}\)): Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant (\(\frac{3\pi}{2}\) to \(2\pi\)): Cosine is positive, and sine and tangent are negative.

Knowing in which quadrant an angle lies helps quickly determine the sign of each trigonometric function without having to calculate their specific values for every angle.