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Chapter 1: Problem 17

Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians. $$\tan (-3 \pi / 4)$$

Short Answer

Expert verified
Answer: The value of $$\tan (-3 \pi / 4)$$ is 1.

Step by step solution

01

Determine the reference angle

To find the reference angle for $$-3 \pi / 4$$, recognize that it is a negative angle, so we'll go clockwise around the unit circle. The positive angle equivalent to $$-3 \pi / 4$$ can be expressed as: $$(-3 \pi / 4) + 2 \pi = (-3 \pi / 4) + 8 \pi / 4 = 5 \pi / 4$$ Now, let's find the reference angle for $$5 \pi / 4$$. We know that $$5 \pi / 4$$ is in the third quadrant, and the reference angle is equal to $$\theta - \pi$$: $$\theta = 5 \pi / 4 - \pi = 5 \pi / 4 - 4 \pi / 4 = \pi / 4$$
02

Find the coordinates on the unit circle

Since the reference angle is $$\pi / 4$$, the coordinates on the unit circle where the terminal side of $$-3 \pi / 4$$ intersects are: $$x = \cos (\pi / 4) = \frac{\sqrt{2}}{2}$$ $$y = \sin (\pi / 4) = \frac{\sqrt{2}}{2}$$ As $$-3 \pi / 4$$ is in the third quadrant, both the x and y coordinates are negative: $$(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$$
03

Compute the tangent

Now, we can calculate the tangent function using the coordinates: $$\tan (-3 \pi / 4) = \frac{y}{x} = \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1$$ So, $$\tan (-3 \pi / 4) = 1$$.
04

Check the result with a calculator

Using a calculator, we can check the result. Ensure that the calculator is in radian mode and compute: $$\tan (-3 \pi / 4)$$ The calculator should also give the result $$1$$, which confirms our manual calculation.

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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 fundamental tool in trigonometry. It is a circle with a radius of 1 unit centered at the origin of the coordinate plane. The significance of the unit circle lies in its powerful relationship with trigonometric functions.

When we wish to evaluate trigonometric functions, like \(\tan (-3 \pi / 4)\) from our textbook exercise, we interpret the angle in terms of a point on the unit circle. This point has coordinates \( (x, y) \) that represent the cosine and sine of the angle, respectively. For an angle in radians, each point on the unit circle outlines the value of sine and cosine at that angle. Since the radius is 1, this simplifies many calculations.
Reference Angle
The concept of a reference angle is incredibly useful in trigonometry. A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It effectively allows us to find the trigonometric functions of any angle by comparing it with a corresponding angle in the first quadrant, where all trigonometric functions are positive.

Our example involves finding the reference angle for \( -3 \pi / 4 \), which resides in the third quadrant. The reference angle for any angle in the third quadrant can be found by subtracting \( \pi \) from the absolute value of the angle, since \( \pi \) radians represents the halfway mark around the circle. Hence, \( \pi / 4 \) is the reference angle for \( -3 \pi / 4 \).
Trigonometric Identities
Trigonometric identities are like the Swiss Army knife for mathematicians working with trigonometric functions鈥攖hey equip them with tools to simplify complex problems. These identities are equations that hold true for all values of the variable when both sides of the equation are defined.

In evaluating the tangent of an angle, such as \( \tan (-3 \pi / 4) \), we use the fundamental identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), where \( \theta \) is the angle of interest. In this scenario, we use the coordinates on the unit circle that correspond to the angle's sine and cosine to calculate the tangent. This relationship shows us the interconnectedness of trigonometric functions and underscores the importance of identities in simplifying trigonometry problems.
Radian Measure
Radians provide us with a way of measuring angles based on the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. Understanding radian measure is vital to solving trigonometry problems because most trigonometric functions are naturally related to a circle's radius.

In the exercise \( \tan (-3 \pi / 4) \), the angle is given in radians, as opposed to degrees, which is another common way of measuring angles. This is significant because the unit circle's framework is inherently based on radian measures. Therefore, working with radians often simplifies calculations and leads to more straightforward interpretations when evaluating trigonometric functions.

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Most popular questions from this chapter

Consider the quartic polynomial \(y=f(x)=x^{4}-x^{2}\) a. Graph \(f\) and estimate the largest intervals on which it is oneto-one. The goal is to find the inverse function on each of these intervals. b. Make the substitution \(u=x^{2}\) to solve the equation \(y=f(x)\) for \(x\) in terms of \(y .\) Be sure you have included all possible solutions. c. Write each inverse function in the form \(y=f^{-1}(x)\) for each of the intervals found in part (a).

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{3} \text { and } y=x^{7}$$

Parabola vertex property Prove that if a parabola crosses the \(x\) -axis twice, the \(x\) -coordinate of the vertex of the parabola is halfway between the \(x\) -intercepts.

Relative acuity of the human eye The fovea centralis (or fovea) is responsible for the sharp central vision that humans use for reading and other detail- oriented eyesight. The relative acuity of a human eye, which measures the sharpness of vision, is modeled by the function $$R(\theta)=\frac{0.568}{0.331|\theta|+0.568}$$ ,where \(\theta\) (in degrees) is the angular deviation of the line of sight from the center of the fovea (see figure). a. Graph \(R,\) for \(-15 \leq \theta \leq 15\) b. For what value of \(\theta\) is \(R\) maximized? What does this fact indicate about our eyesight? c. For what values of \(\theta\) do we maintain at least \(90 \%\) of our maximum relative acuity? (Source: The Journal of Experimental Biology, 203, Dec 2000)

Viewing angles An auditorium with a flat floor has a large flat panel television on one wall. The lower edge of the television is \(3 \mathrm{ft}\) above the floor, and the upper edge is \(10 \mathrm{ft}\) above the floor (see figure). Express \(\theta\) in terms of \(x.\)
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