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7.E: Trigonometric Identities and Equations (Exercises)

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7.1: Solving Trigonometric Equations with Identities

In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions.

7.2: Sum and Difference Identities

In this section, we will learn techniques that will enable us to solve useful problems. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the termformula is used synonymously with the word identity.

7.3: Double-Angle, Half-Angle, and Reduction Formulas

In this section, we will investigate three additional categories of identities. Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not.

7.4: Sum-to-Product and Product-to-Sum Formulas

From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine. The product-to-sum formulas can rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. We can also derive the sum-to-product identities from the product-to-sum identities using substitution. The sum-to-product formulas are used to rewrite sum or difference as products of sines and cosines.

7.5: Solving Trigonometric Equations

In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.

7.6: Modeling with Trigonometric Equations

Many natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year. So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern and alter the function to get adependable model. Here. we will take a deeper look at specific types of periodic behavior and model equations to fit data.

Section Exercises

Verbal

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

Physical behavior should be periodic, or cyclical.

What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

Explain the effect of a damping factor on the graphs of harmonic motion functions.

Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

 $$x$$ $$y$$ $$0$$ $$−4$$ $$3$$ $$−1$$ $$6$$ $$2$$ $$9$$ $$−1$$ $$12$$ $$−4$$ $$15$$ $$−1$$ $$18$$ $$2$$

$$y=−3 \cos (\frac{π}{6}x)−1$$

 $$x$$ $$y$$ $$0$$ $$5$$ $$2$$ $$1$$ $$4$$ $$−3$$ $$6$$ $$1$$ $$8$$ $$5$$ $$10$$ $$1$$ $$12$$ $$−3$$
 $$x$$ $$y$$ $$0$$ $$2$$ $$\frac{π}{4}$$ $$7$$ $$\frac{π}{2}$$ $$2$$ $$\frac{3π}{4}$$ $$−3$$ $$π$$ $$2$$ $$\frac{5π}{4}$$ $$7$$ $$\frac{3π}{2}$$ $$2$$

$$5 \sin (2x)+2$$

 $$x$$ $$y$$ $$0$$ $$2$$ $$\frac{π}{4}$$ $$7$$ $$\frac{π}{2}$$ $$2$$ $$\frac{3π}{4}$$ $$−3$$ $$π$$ $$2$$ $$\frac{5π}{4}$$ $$7$$ $$\frac{3π}{2}$$ $$2$$
 $$x$$ $$y$$ $$0$$ $$1$$ $$1$$ $$−3$$ $$2$$ $$−7$$ $$3$$ $$−3$$ $$4$$ $$1$$ $$5$$ $$−3$$ $$6$$ $$−7$$

$$4 \cos (\frac{xπ}{2})−3$$

 $$x$$ $$y$$ $$0$$ $$−2$$ $$1$$ $$4$$ $$2$$ $$10$$ $$3$$ $$4$$ $$4$$ $$−2$$ $$5$$ $$4$$ $$6$$ $$10$$
 $$x$$ $$y$$ $$0$$ $$5$$ $$1$$ $$−3$$ $$2$$ $$5$$ $$3$$ $$13$$ $$4$$ $$5$$ $$5$$ $$−3$$ $$6$$ $$5$$

$$5−8 \sin (\frac{xπ}{2})$$

 $$x$$ $$y$$ $$−3$$ $$−1−\sqrt{2}$$ $$−2$$ $$−1$$ $$−1$$ $$1−\sqrt{2}$$ $$0$$ $$0$$ $$1$$ $$\sqrt{2}−1$$ $$2$$ $$1$$ $$3$$ $$\sqrt{2}+1$$
 $$x$$ $$y$$ $$−1$$ $$\sqrt{3}−2$$ $$0$$ $$0$$ $$1$$ $$2−\sqrt{3}$$ $$2$$ $$\frac{\sqrt{3}}{3}$$ $$3$$ $$1$$ $$4$$ $$\sqrt{3}$$ $$5$$ $$2+\sqrt{3}$$

$$\tan (\frac{xπ}{12})$$

Graphical

For the following exercises, graph the given function, and then find a possible physical process that the equation could model.

$$f(x)=−30 \cos (\frac{xπ}{6})−20 \cos ^2 (\frac{xπ}{6})+80 \; [0,12]$$

$$f(x)=−18 \cos (\frac{xπ}{12})−5 \sin (\frac{xπ}{12})+100$$ on the interval $$[0,24]$$

Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.

$$f(x)=10−\sin (\frac{xπ}{6})+24 \tan (\frac{xπ}{240})$$ on the interval $$[0,80]$$

Technology

For the following exercise, construct a function modeling behavior and use a calculator to find desired results.

A city’s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below 0.

9 years from now

Real-World Applications

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $$105°F$$ occurs at 5PM and the average temperature for the day is $$85°F$$. Find the temperature, to the nearest degree, at 9AM.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $$84°F$$ occurs at 6PM and the average temperature for the day is $$70°F.$$ Find the temperature, to the nearest degree, at 7AM.

$$56 °F$$

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $$47°F$$ and $$63°F$$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $$51°F$$?

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $$64°F$$ and $$86°F$$ during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach $$70°F$$?

$$1.8024$$ hours

A Ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How much of the ride, in minutes and seconds, is spent higher than 13 meters above the ground?

A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second

4:30

The sea ice area around the North Pole fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.

The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.

From July 8 to October 23

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between a low of 2 inches on day 10 and 12 inches on day 55, during what period is daily rainfall more than 10 inches?

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?

From day 19 through day 40

In a certain region, monthly precipitation peaks at 8 inches on June 1 and falls to a low of 1 inch on December 1. Identify the periods when the region is under flood conditions (greater than 7 inches) and drought conditions (less than 2 inches). Give your answer in terms of the nearest day.

In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.

Floods: July 24 through October 7. Droughts: February 4 through March 27

For the following exercises, find the amplitude, period, and frequency of the given function.

The displacement $$h(t)$$ in centimeters of a mass suspended by a spring is modeled by the function $$h(t)=8 \sin (6πt),$$ where $$t$$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

The displacement $$h(t)$$ in centimeters of a mass suspended by a spring is modeled by the function $$h(t)=11 \sin (12πt),$$ where $$t$$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Amplitude: 11, period: $$\frac{1}{6}$$, frequency: 6 Hz

The displacement $$h(t)$$ in centimeters of a mass suspended by a spring is modeled by the function $$h(t)=4 \cos (\frac{π}{2}t)$$, where $$t$$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

The displacement $$h(t)$$, in centimeters, of a mass suspended by a spring is modeled by the function $$h(t)=−5 \cos (60πt)$$, where $$t$$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Amplitude: 5, period: $$\frac{1}{30}$$, frequency: 30 Hz

For the following exercises, construct an equation that models the described behavior.

A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, $$P$$, in terms of months since January, $$t$$.

A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, $$P$$, in terms of months since January, $$t$$.

$$P(t)=−15 \cos (\frac{π}{6}t)+650+\frac{55}{6}t$$

A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, $$P$$, in terms of months since January, $$t$$.

A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, $$P$$, in terms of months since January, $$t$$.

$$P(t)=−40 \cos (\frac{π}{6}t)+800(1.04)^t$$

A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, $$D$$, the end of the spring is from equilibrium in terms of seconds, $$t$$, since the spring was released.

A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, $$D$$, the end of the spring is from equilibrium in terms of seconds, $$t,$$ since the spring was released.

$$D(t)=7(0.89)^t \cos (40πt)$$

A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, $$D,$$ the end of the spring is from equilibrium in terms of seconds, $$t$$, since the spring was released.

A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, $$D$$, the end of the spring is from equilibrium in terms of seconds, $$t$$, since the spring was released.

$$D(t)=19(0.9265)^t \cos (26πt)$$

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?

Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?

$$20.1$$ years

A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between −0.1 and 0.1 cm,effectively at rest.

A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between −0.1 and 0.1 cm,effectively at rest.

17.8 seconds

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 8 times per second, was initially pulled down 32 cm from equilibrium, and the amplitude decreases by 50% each second. The second spring, oscillating 18 times per second, was initially pulled down 15 cm from equilibrium and after 4 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than 0.1 cm.

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than 0.1 cm.

Spring 2 comes to rest first after 8.0 seconds.

Extensions

A plane flies 1 hour at 150 mph at $$22^∘$$ east of north, then continues to fly for 1.5 hours at 120 mph, this time at a bearing of $$112^∘$$ east of north. Find the total distance from the starting point and the direct angle flown north of east.

A plane flies 2 hours at 200 mph at a bearing of $$60^∘$$,then continues to fly for 1.5 hours at the same speed, this time at a bearing of $$150^∘$$. Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.

500 miles, at $$90^∘$$

For the following exercises, find a function of the form y=abx+csin(π2x) y=abx+csin(π2x) that fits the given data.

 $$x$$ 0 1 2 3 $$y$$ 6 29 96 379
 $$x$$ 0 1 2 3 $$y$$ 6 34 150 746

$$y=6(5)^x+4 \sin (\frac{π}{2}x)$$

 $$x$$ 0 1 2 3 $$y$$ 4 0 16 -40

For the following exercises, find a function of the form $$y=ab^x \cos (\frac{π}{2}x)+c$$ that fits the given data.

 $$x$$ 0 1 2 3 $$y$$ 11 3 1 3

$$y=8(\frac{1}{2})^x \cos (\frac{π}{2}x)+3$$

 $$x$$ 0 1 2 3 $$y$$ 4 1 −11 1