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Chapter 5: Problem 4

Use graphing technology to graph \(y=\) tan \(x\) using the following window settings: \(x:\left[-360^{\circ}, 360^{\circ}, 30^{\circ}\right]\) and \(y:[-3,3,1] .\) Trace along the graph to locate the value of tan \(x\) when \(x=60^{\circ}\) Predict the other values of \(x\) that will produce the same value for tan \(x\) within the given domain. Verify your predictions.

Short Answer

Expert verified
The values of x that produce the same value for \(\tan(x)\) as 60掳 within the given domain are 240掳, -120掳, and -300掳.

Step by step solution

01

Set Up Graphing Technology

Use a graphing calculator or graphing software. Set the x-axis window to \([-360^{\circ}, 360^{\circ}, 30^{\circ}]\) and the y-axis window to \([-3, 3, 1]\).
02

Graph the Function

Enter the function \(\tan(x)\) into the graphing tool and create the graph.
03

Trace the Graph

Use the trace feature of your graphing tool to move along the graph and locate the value of \(\tan(x)\) at \({x=60^{\circ}}\).
04

Record the Value

Note down the value of \(\tan(60^{\circ})\). It should be \(\tan(60^{\circ}) = \sqrt{3}\) or approximately 1.732.
05

Predict Other Values

Observe the periodicity of the tangent function, which repeats every \({180^{\circ}}\). Predict that \(\tan(x) = \tan(x + 180^{\circ}k)\) where \({k\)} is an integer.
06

List the Predicted Values

Within \([-360^{\circ}, 360^{\circ}]\), the predicted values for \({x}\) are \(\tan(60^{\circ}), \tan(240^{\circ}), \tan(-120^{\circ})\) and \(\tan(-300^{\circ})\).
07

Verify Predictions

Use the trace feature to check the values of \(\tan(x)\) at \({240^{\circ}}, {-120^{\circ}}, \text{and} {-300^{\circ}}\). All should give the same value.

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Key Concepts

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

periodicity of tangent function
The tangent function is periodic, meaning it repeats its values at regular intervals. For tangent functions, this interval is every \(180^\text{掳}\) or \(\frac{\text{蟺}}{\text{蟺}}\) in radians. This characteristic makes it easier to predict the behavior of the function within a given domain. For example, if \(\tan(\text{60}^\text{掳})\) is \( \sqrt{3}\) around 1.732, then \tan(240^\text{掳})\, \tan(-120^\text{掳})\ and \tan(-300^\text{掳})\text{ will also yield the same values}\text{. Knowing this periodic nature helps in predicting and verifying the values of} tangential\text{functions over various intervals}\
trigonometric functions
Trigonometric functions like sine, cosine, and tangent are essential in understanding wave patterns, oscillations, and circular motion. The tangent function, specifically, relates the angle in a right triangle to the ratio of the opposite side over the adjacent side. The general form of the tangent function is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) The tangent function is unique because it has vertical asymptotes where the function is undefined, occurring every \( \theta = \frac{\text{蟺}}{\text{2}} + k\text{蟺}\)\text{where}k\text{ is an integer}. These are the points where the function's values surge to positive or negative infinity. Understanding these properties is crucial for analyzing and interpreting graphs of trigonometric functions.
using graphing technology
Using graphing technology can significantly enhance our understanding of mathematical functions and their behaviors. Tools like graphing calculators or software allow us to visually explore functions such as \( \tan(x)\). By setting appropriate window settings for \(x\) and \(y\) values, we can observe the function over a specified interval. The window setting of \( \left[\text{-360}^\text{掳}, \text{ 360}^\text{掳}, \text{30}^\text{掳}\right]\) for the \( x \)-axis and \( \left[\text{-3,3,1}\text{y}\right]\) allows us to see multiple periods and behaviors of the function. Using the trace feature of these tools helps in finding specific values, verifying periodicity, and enhancing predictive analysis. By utilizing these technologies effectively, students can visualize and understand complex mathematical concepts better and build strong analytical skills.

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Most popular questions from this chapter

a) Copy each equation. Fill in the missing values to make the equation true. i) \(4 \sin \left(x-30^{\circ}\right)=4 \cos (x-\square)\) ii) \(2 \sin \left(x-\frac{\pi}{4}\right)=2 \cos (x-\square)\) iii) \(-3 \cos \left(x-\frac{\pi}{2}\right)=3 \sin (x+\square)\) iv) \(\cos (-2 x+6 \pi)=\sin 2(x+\square)\) b) Choose one of the equations in part a and explain how you got your answer.

Point \(\mathrm{P}(x, y)\) is plotted where the terminal arm of angle \(\theta\) intersects the unit circle. a) Use \(\mathrm{P}(x, y)\) to determine the slope of the terminal arm. b) Explain how your result from part a) is related to tan \(\theta\) c) Write your results for the slope from part a) in terms of sine and cosine. d) From your answer in part c), explain how you could determine tan \(\theta\) when the coordinates of point \(P\) are known.

The typical voltage, \(V\), in volts (V), supplied by an electrical outlet in Cuba is a sinusoidal function that oscillates between \(-155 \mathrm{V}\) and \(+155 \mathrm{V}\) and makes 60 complete cycles each second. Determine an equation for the voltage as a function of time, \(t.\)

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)

After exercising for 5 min, a person has a respiratory cycle for which the rate of air flow, \(r,\) in litres per second, in the lungs is approximated by \(r=1.75 \sin \frac{\pi}{2} t,\) where \(t\) is the time, in seconds. a) Determine the time for one full respiratory cycle. b) Determine the number of cycles per minute. c) Sketch the graph of the rate of air flow function. d) Determine the rate of air flow at a time of 30 s. Interpret this answer in the context of the respiratory cycle. e) Determine the rate of air flow at a time of 7.5 s. Interpret this answer in the context of the respiratory cycle.
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