Problem 24 - Contemporary Precalculus

Question

In Exercises \(21-25,\) evaluate all six trigonometric finctions at the given number without using a calculator. $$\frac{11 \pi}{3}$$

Step-by-Step Solution

Answer

Question: Determine the six trigonometric functions for the angle \(\frac{11 \pi}{3}\) without using a calculator. Answer: 1. \(\sin{\frac{11 \pi}{3}} = -\frac{\sqrt{3}}{2}\) 2. \(\cos{\frac{11 \pi}{3}} = \frac{1}{2}\) 3. \(\tan{\frac{11 \pi}{3}} = -\frac{\sqrt{3}}{3}\) 4. \(\csc{\frac{11 \pi}{3}} = -\frac{2}{\sqrt{3}}\) 5. \(\sec{\frac{11 \pi}{3}} = 2\) 6. \(\cot{\frac{11 \pi}{3}} = -\frac{\sqrt{3}}{3}\)

Step 1: Find a Co-terminal Angle

To find a co-terminal angle within the range of \(0\) to \(2 \pi\), we can add or subtract any multiple of \(2 \pi\) to the given angle. In this case, subtracting \(4 \pi\) (which is \(2 \times 2 \pi\)) from the given angle will give us a co-terminal angle within the desired range: $$\frac{11 \pi}{3} - 4 \pi = \frac{11 \pi}{3} - \frac{12 \pi}{3} = -\frac{\pi}{3}$$

Step 2: Determine Quadrant and Reference Angle

The co-terminal angle we calculated is negative, which means it is in the opposite direction (clockwise). To make it easier, we will find the positive co-terminal angle, which will be in the counter-clockwise direction, and then determine the quadrant it lies in: $$\frac{11 \pi}{3} - 2 \pi$$ The positive co-terminal angle will be: $$2 \pi - \frac{\pi}{3} = \frac{6 \pi}{3} - \frac{\pi}{3} = \frac{5 \pi}{3}$$ The angle \(\frac{5 \pi}{3}\) lies in the fourth quadrant, since it is between \(\frac{4 \pi}{3}\) and \(2 \pi\).

Step 3: Find Trigonometric Functions

In the fourth quadrant, sine and tangent functions are negative, while cosine is positive. The reference angle (the angle formed with the positive x-axis) is \(\pi - \frac{5 \pi}{3} = \frac{2 \pi}{3}\). We can now find the six trigonometric functions using either reference angles or known angles with specific values: 1. \(\sin{\frac{11 \pi}{3}} = -\sin{\frac{2 \pi}{3}} = -\frac{\sqrt{3}}{2}\) 2. \(\cos{\frac{11 \pi}{3}} = \cos{\frac{5 \pi}{3}} = \cos{(-\frac{\pi}{3})} = \frac{1}{2}\) 3. \(\tan{\frac{11 \pi}{3}} = -\tan{\frac{2 \pi}{3}} = -\frac{\sqrt{3}}{3}\) 4. \(\csc{\frac{11 \pi}{3}} = -\csc{\frac{2 \pi}{3}} = -\frac{2}{\sqrt{3}}\) 5. \(\sec{\frac{11 \pi}{3}} = \sec{\frac{5 \pi}{3}} = \sec{(-\frac{\pi}{3})} = 2\) 6. \(\cot{\frac{11 \pi}{3}} = -\cot{\frac{2 \pi}{3}} = -\frac{\sqrt{3}}{3}\) Note that we used the reference angle \(\frac{2 \pi}{3}\) (or \(120^{\circ}\)) and its complementary angle in the fourth quadrant \(-\frac{\pi}{3}\) (or \(-60^{\circ}\)) to find the trigonometric functions. We also considered the signs of these functions in the fourth quadrant, where sine and tangent are negative, and cosine is positive.

Key Concepts

Co-Terminal Angles

Co-terminal angles are angles that share the same terminal side when drawn in standard position (with their initial side along the positive x-axis). These angles differ by multiples of full rotations, or increments of \(2\pi\) radians (360 degrees). When you're given an angle such as \(\frac{11\pi}{3}\), which is outside the usual range of \(0\) to \(2\pi\), finding a co-terminal angle within this range simplifies the task of evaluating trigonometric functions.

For instance, to find a co-terminal angle for \(\frac{11\pi}{3}\), you can subtract \(4\pi\) (\(\frac{12\pi}{3}\)) to get \(-\frac{\pi}{3}\). However, because it's often easier to work with positive angles, you can add \(2\pi\) to find the positive co-terminal angle, which gives \(\frac{5\pi}{3}\). By understanding how to manipulate angles to find their co-terminals, you can greatly simplify the process of evaluating trigonometric functions without the aid of a calculator.

Reference Angles

Reference angles help us find trigonometric function values quickly, especially for angles that are not an immediate part of the unit circle's familiar points. A reference angle is formed by the terminal side of an angle and the x-axis. Essentially, it’s the smallest angle that the terminal side makes with the x-axis.

For an angle like \(\frac{5\pi}{3}\), which lies in the fourth quadrant, the reference angle would be \(2\pi - \frac{5\pi}{3} = \frac{\pi}{3}\). This positive acute angle is your key to finding trigonometric values. Since the reference angle \(\frac{\pi}{3}\) has known sine, cosine, and tangent values, you can use them directly, adjusting for the sign based on the original angle's quadrant. This method avoids the need for memorization of numerous trig values and is particularly effective for finding trigonometric functions without a calculator.

Trigonometric Functions Without A Calculator

Evaluating trigonometric functions without a calculator relies on a good grasp of key concepts such as co-terminal and reference angles. Once you've found a co-terminal angle within \(0\) to \(2\pi\) and its corresponding reference angle, you can determine the signs and values of the six trigonometric functions based on the quadrant in which the angle lies.

In our example, since \(\frac{5\pi}{3}\) is in the fourth quadrant, we know that the sine and tangent functions will have negative values while the cosine function will be positive. Using the reference angle, we can directly apply the known trigonometric values for \(\frac{\pi}{3}\), adjusting for the signs pertinent to the fourth quadrant. This approach allows you to find accurate values for the sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) of any given angle, even without a calculator, harnessing the simplicity of reference angles alongside the unique properties of each quadrant.