Problem 25 - Contemporary Precalculus

Question

In Exercises \(21-25,\) evaluate all six trigonometric finctions at the given number without using a calculator. $$ \frac{-11 \pi}{4} $$

Step-by-Step Solution

Answer

The values for all six trigonometric functions for the angle \(\frac{-11 \pi}{4}\) are: - \(\sin\left(\frac{-11 \pi}{4}\right) = -\frac{\sqrt{2}}{2}\) - \(\cos\left(\frac{-11 \pi}{4}\right) = -\frac{\sqrt{2}}{2}\) - \(\tan\left(\frac{-11 \pi}{4}\right) = 1\) - \(\csc\left(\frac{-11 \pi}{4}\right) = -\sqrt{2}\) - \(\sec\left(\frac{-11 \pi}{4}\right) = -\sqrt{2}\) - \(\cot\left(\frac{-11 \pi}{4}\right) = 1\)

Step 1: Find a coterminal angle

Given an angle \(\frac{-11 \pi}{4}\), we can find a coterminal angle between \(0\) and \(2 \pi\) by adding multiples of \(2 \pi\). In this case, multiplying \(2 \pi\) by \(3\) gives us \(6 \pi\). Now, add this value to the given angle: $$ \frac{-11 \pi}{4} + 6 \pi = \frac{-11 \pi + 24 \pi}{4} = \frac{13 \pi}{4} $$ The coterminal angle is \(\frac{13 \pi}{4}\), which is greater than \(2 \pi\). To get an angle within this range, we can subtract \(2 \pi\) from the angle to find another coterminal angle: $$ \frac{13 \pi}{4} - 2 \pi = \frac{13 \pi - 8 \pi}{4} = \frac{5 \pi}{4} $$ This coterminal angle value (\(\frac{5 \pi}{4}\)) is within the range of \(0\) to \(2 \pi\).

Step 2: Determine sine and cosine values

To find the sine and cosine values for the coterminal angle \(\frac{5 \pi}{4}\), consider the position of the point on the unit circle. At this angle, sine and cosine (which are the coordinates of the point on the unit circle) are equivalent. Since \(\frac{5 \pi}{4}\) is in the third quadrant, both sine and cosine will be negative. Thus, we have: $$ \sin\left(\frac{5 \pi}{4}\right) = \cos\left(\frac{5 \pi}{4}\right) = -\frac{\sqrt{2}}{2} $$

Step 3: Find tangent value

To find the tangent value for the coterminal angle \(\frac{5 \pi}{4}\), use the following definition: $$ \tan\left(\frac{5 \pi}{4}\right) = \frac{\sin\left(\frac{5 \pi}{4}\right)}{\cos\left(\frac{5 \pi}{4}\right)} $$ Substitute the sine and cosine values we found in step 2 and simplify: $$ \tan\left(\frac{5 \pi}{4}\right) = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $$

Step 4: Determine cosecant, secant, and cotangent values

Now we can find the values for cosecant, secant, and cotangent using the reciprocal definitions: $$ \csc\left(\frac{5 \pi}{4}\right) = \frac{1}{\sin\left(\frac{5 \pi}{4}\right)} = -\sqrt{2} $$ $$ \sec\left(\frac{5 \pi}{4}\right) = \frac{1}{\cos\left(\frac{5 \pi}{4}\right)} = -\sqrt{2} $$ $$ \cot\left(\frac{5 \pi}{4}\right) = \frac{1}{\tan\left(\frac{5 \pi}{4}\right)} = 1 $$

Step 5: Final answers

The trigonometric function values for the given angle are: $$ \sin\left(\frac{-11 \pi}{4}\right) = -\frac{\sqrt{2}}{2} $$ $$ \cos\left(\frac{-11 \pi}{4}\right) = -\frac{\sqrt{2}}{2} $$ $$ \tan\left(\frac{-11 \pi}{4}\right) = 1 $$ $$ \csc\left(\frac{-11 \pi}{4}\right) = -\sqrt{2} $$ $$ \sec\left(\frac{-11 \pi}{4}\right) = -\sqrt{2} $$ $$ \cot\left(\frac{-11 \pi}{4}\right) = 1 $$

Key Concepts

Coterminal Angles

Understanding coterminal angles is critical in trigonometry. These are angles that share the same initial and terminal sides but differ in how many full rotations they encompass. To find a coterminal angle, add or subtract multiples of the full angle measure of a circle, which is \(2 \pi\) radians or 360 degrees.

For the exercise involving \(\frac{-11 \pi}{4}\), converting it to a coterminal angle between 0 and \(2 \pi\) helps us evaluate its trigonometric functions more straightforwardly. Here's how it’s done: by adding \(6 \pi\), we initially overshoot into \(\frac{13 \pi}{4}\), and then by subtracting \(2 \pi\), we get our coterminal angle \(\frac{5 \pi}{4}\) within the desired range. Coterminal angles let us easily map angles onto the unit circle and support computation of trigonometric functions without reaching for a calculator.

Unit Circle

Mapping Angles onto the Circle

The unit circle is a powerful tool in trigonometry, representing all possible positions of an angle as points along a circle with a radius of one. This circle is centered at the origin of the coordinate plane. The angle values are measured in radians from the positive x-axis (0 radians) counter-clockwise.

Considering the coterminal angle of \(\frac{5 \pi}{4}\), we find its position in the third quadrant of the unit circle, where both x (cosine) and y (sine) coordinates are negative. This specific angle corresponds to a coordinate with equal x and y values, providing the sine and cosine values of \( -\frac{\sqrt{2}}{2} \).

Why It's Useful

The unit circle simplifies the process of finding trigonometric values for angles beyond the first revolution or angles expressed in a negative direction, ensuring consistency and ease of understanding in trigonometric evaluations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions give us additional perspectives on angles. They include cosecant (csc), secant (sec), and cotangent (cot), which are the reciprocals of sine, cosine, and tangent respectively. They can be expressed as:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

For instance, to find these functions for the coterminal angle \(\frac{5 \pi}{4}\), we use the previously found sine and cosine values. The results reflect the third quadrant's property, having both csc and sec remaining negative due to the negative sine and cosine values. Cotangent, being the reciprocal of tangent, which is 1 in this case, remains unaffected. These reciprocal relationships allow for a fuller understanding and seamless calculation of all trigonometric functions.