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Chapter 1: Problem 19

Evaluate the following expressions by drawing the unit circle and the appropriate right triangle. Use a calculator only to check your work. All angles are in radians. $$\cot (-13 \pi / 3)$$

Short Answer

Expert verified
Answer: cot(-13蟺/3) = 1/鈭�3

Step by step solution

01

Find the reference angle and its position on the unit circle

To find the reference angle, we need to express the angle \((-13\pi/3)\) in a range of \([0, 2\pi)\). This can be done by adding multiples of \(2\pi\) to the angle until it falls within this range. $$(-13\pi/3) + 8\pi = (-13\pi/3) + (24\pi/3) = 11\pi/3.$$ Now we need to find which quadrant the angle is in: $$11\pi/3 = 3\pi + 2\pi/3$$ The angle \(11\pi/3\) is in the fourth quadrant and has a reference angle of \(2\pi/3\).
02

Construct the right triangle

Draw a unit circle and locate the point where the angle \(11\pi/3\) intersects the circle. Since \(11\pi/3\) is in the fourth quadrant, its right triangle will have its adjacent side on the positive x-axis and the opposite side going downwards on the negative y-axis. Using the reference angle \(2\pi/3\), we can find the coordinates of the point of intersection (P) between the terminal side of the angle and the unit circle using sine and cosine functions. $$x = \cos(2\pi/3) = -1/2$$ $$y = \sin(2\pi/3) = \sqrt{3}/2$$ Since the angle is in the fourth quadrant, the y-coordinate will be negative. So, the coordinates of point P are: $$P = (-1/2, -\sqrt{3}/2)$$
03

Use the cotangent function to find the value of \(\cot(-13\pi/3)\)

Recall that the cotangent function is defined as the ratio of the adjacent side to the opposite side. In our case, the adjacent side has a length of \(1/2\) and the opposite side has a length of \(\sqrt{3}/2\). So, the cotangent function for the angle \(11\pi/3\) (which is equivalent to \(-13\pi/3\)) is: $$\cot(-13 \pi / 3) = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$$ The expression for \(\cot(-13\pi/3)\) is: $$\cot(-13\pi/3) = \frac{1}{\sqrt{3}}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a key tool in trigonometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
By using this circle, we can easily understand and visualize the trigonometric functions.
  • Each point on the unit circle corresponds to an angle in radians from the positive x-axis.
  • The coordinates of each point are in the form \( (\cos(\theta), \sin(\theta)) \).
This circle helps us determine the values of sine and cosine for any given angle. From these values, we can also find other trigonometric functions like tangent and cotangent. Remember:
  • The circle allows us to visualize angles beyond \(2\pi\) by wrapping them around the circle.
  • Negative angles are measured clockwise from the positive x-axis.
Cotangent Function
The cotangent function gives the ratio of the adjacent side to the opposite side in a right triangle, which can be built using the coordinates of the unit circle.
It is defined as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).
  • In terms of the unit circle, cotangent uses the x-coordinate (cosine) and the y-coordinate (sine).
  • Cotangent is undefined when \(\sin(\theta) = 0\) because division by zero is not possible.
To evaluate \( \cot(-13\pi/3) \), we first find the equivalent angle within \(2\pi\). By using the reference angle and the coordinates on the unit circle, we calculate the cotangent ratio.
  • With \( P = (-1/2, -\sqrt{3}/2) \), the cotangent is \( \frac{-1/2}{-\sqrt{3}/2} = \frac{1}{\sqrt{3}} \).
This illustrates how cotangent depends on the angle's position within the unit circle.
Reference Angle
Reference angles are useful in finding angles' positions on the unit circle. They help us determine the angle's value in terms of commonly known angles like \(\pi/3\) or \(\pi/4\).
Here's what you need to know:
  • A reference angle is the acute angle an angle makes with the x-axis.
  • It can always be found between 0 and \(\pi/2\), or between 0 and \(90^\circ\) for degree measures.
For the angle \( -13\pi/3 \), we simplify it within \(2\pi\) to find \( 11\pi/3 \). The reference angle tells us the equivalent angle in its quadrant.
  • Here, that reference angle is \( 2\pi/3 \), located in the fourth quadrant.
Understanding reference angles helps in predicting function signs and values across different quadrants on the unit circle.

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Most popular questions from this chapter

The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles; one by wrapping a rope around the equator and then a second circle that is \(38 \mathrm{ft}\) longer than the first rope (see figure). How much space is between the ropes?

Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility only to check your work. $$p(x)=3 \sin (2 x-\pi / 3)+1$$

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabolas \(y=x^{2}\) and \(y=-x^{2}+8 x\)

Modify Exercise 84 and use property \(\mathrm{E} 2\) for exponents to prove that \(\log _{b}(x / y)=\log _{b} x-\log _{b} y\).

Use the definition of absolute value to graph the equation \(|x|-|y|=1 .\) Use a graphing utility only to check your work.
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