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Mathematics LibreTexts

1.7.R: Trigonometric Identities and Equations (Review)

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7.1: Solving Trigonometric Equations with Identities

For the exercises 1-6, find all solutions exactly that exist on the interval [0,2π).

1) csc2t=3

Answer

sin1(33),πsin1(33),π+sin1(33),2πsin1(33)

2) cos2x=14

3) 2sinθ=1

Answer

7π6,11π6

4) tanxsinx+sin(x)=0

5) 9sinω2=4sin2ω

Answer

sin1(14),πsin1(14)

6) 12tan(ω)=tan2(ω)

For the exercises 7-8, use basic identities to simplify the expression.

7) secxcosx+cosx1secx

Answer

1

8) sin3x+cos2xsinx

For the exercises 9-10, determine if the given identities are equivalent.

9) sin2x+sec2x1=(1cos2x)(1+cos2x)cos2x

Answer

Yes

10) tan3xcsc2xcot2xcosxsinx=1

7.2: Sum and Difference Identities

For the exercises 1-4, find the exact value.

1) tan(7π12)

Answer

23

2) cos(25π12)

3) sin(70)cos(25)cos(70)sin(25)

Answer

22

4) cos(83)cos(23)+sin(83)sin(23)

For the exercises 5-6, prove the identity.

5) cos(4x)cos(3x)cosx=sin2x4cos2xsin2x

Answer

cos(4x)cos(3x)cosx=cos(2x+2x)cos(x+2x)cosx=cos(2x)cos(2x)sin(2x)sin(2x)cosxcos(2x)cosx+sinxsin(2x)cosx=(cos2xsin2x)24cos2xsin2xcos2x(cos2xsin2x)+sinx(2)sinxcosxcosx=(cos2xsin2x)24cos2xsin2xcos2x(cos2xsin2x)+2sin2xcos2x=cos4x2cos2xsin2x+sin4cos2xsin2xcos4x+cos2xsin2x+2sin2xcos2x=sin4x4cos2xsin2x+cos2xsin2x=sin2x(sin2x+cos2x)4cos2xsin2x=sin2x4cos2xsin2x

6) cos(3x)cos3x=cosxsin2xsinxsin(2x)

For exercise 7, simplify the expression.

7) tan(12x)+tan(18x)1tan(18x)tan(12x)

Answer

tan(58x)

For the exercises 8-9, find the exact value.

8) cos(sin1(0)cos1(12))

9) tan(sin1(0)sin1(12))

Answer

33

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

For the exercises 1-4, find the exact value.

1) Find sin(2θ), cos(2θ), and tan(2θ) given cosθ=13 and θ is in the interval [π2,π].

2) Find sin(2θ), cos(2θ), and tan(2θ) given secθ=53 and θ is in the interval [π2,π].

Answer

2425,725,247

3) sin(7π8)

4) sec(3π8)

Answer

2(2+2)

For the exercises 5-6, use Figure below to find the desired quantities.

CNX_Precalc_Figure_07_07_201.jpg

5) sin(2β),cos(2β),tan(2β),sin(2α),cos(2α),tan(2α)

6) sin(β2),cos(β2),tan(β2),sin(α2),cos(α2),tan(α2)

Answer

210,7210,17,35,45,34

For the exercises 7-8, prove the identity.

7) 2cos(2x)sin(2x)=cotxtanx

8) cotxcos(2x)=sin(2x)+cotx

Answer

cotxcos(2x)=cotx(12sin2x)=cotxcosxsinx(2)sin2x=2sinxcos=sin(2x)+cotx

For the exercises 9-10, rewrite the expression with no powers.

9) cos2xsin4(2x)

10) tan2xsin3x

Answer

10sinx5sin(3x)+sin(5x)8(cos(2x)+1)

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

For the exercises 1-3, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

1) cos(π3)sin(π4)

2) 2sin(2π3)sin(5π6)

Answer

32

3) 2cos(π5)cos(π3)

For the exercises 4-5, evaluate the sum by using a product formula. Write the exact answer.

4) sin(π12)sin(7π12)

Answer

22

5) cos(5π12)+cos(7π12)

For the exercises 6-9, change the functions from a product to a sum or a sum to a product.

6) sin(9x)cos(3x)

Answer

12(sin(6x)+sin(12x))

7) cos(7x)cos(12x)

8) sin(11x)+sin(2x)

Answer

2sin(132x)cos(92x)

9) cos(6x)+cos(5x)

7.5: Solving Trigonometric Equations

For the exercises 1-2, find all exact solutions on the interval [0,2π).

1) tanx+1=0

Answer

3π4,7π4

2) 2sin(2x)+2=0

For the exercises 3-7, find all exact solutions on the interval [0,2π).

3) 2sin2xsinx=0

Answer

0,π6,5π6,π

4) cos2xcosx1=0

5) 2sin2x+5sinx+3=0

Answer

3π2

6) cosx5sin(2x)=0

7) 1sec2x+2+sin2x+4cos2x=0

Answer

No solution.

For the exercises 8-9, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0,2π) .Round to four decimal places.

8) 3cot2x+cotx=1

9) csc2x3cscx4=0

Answer

0.2527,2.8889,4.7124

For the exercises 10-11, graph each side of the equation to find the zeroes on the interval [0,2π).

10) 20cos2x+21cosx+1=0

11) sec2x2secx=15

Answer

1.3694,1.9106,4.3726,4.9137

7.6: Modeling with Trigonometric Equations

For the exercises 1-3, graph the points and find a possible formula for the trigonometric values in the given table.

1)

x 0 1 2 3 4 5
y 1 6 11 6 1 6

2)

x y
0 -2
1 1
2 -2
3 -5
4 -2
5 1
Answer

3sin(xπ2)2

3)

x y
-3 3+22
-2 3
-1 221
0 1
1 322
2 -1
3 122

4) A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

Answer

71.6

5) Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

For the exercises 6-7, construct functions that model the described behavior.

6) A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to t, the month.

Answer

P(t)=950450sin(π6t)

7) Daily temperatures in the desert can be very extreme. If the temperature varies from 90F to 30F and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the exercises 8-9, find the amplitude, frequency, and period of the given equations.

8) y=3cos(xπ)

Answer

Amplitude: 3, period: 2, frequency: 12Hz

9) y=2sin(16xπ)

For the exercises 10-11, model the described behavior and find requested values.

10) An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to t, the number of years from now.

Answer

C(t)=20sin(2πt)+100(1.4427)t

11) The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to t, the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.

Practice Test

For the exercises 1-2, simplify the given expression.

1) cos(x)sinxcotx+sin2x

Answer

1

2) sin(x)cos(2x)sin(x)cos(2x)

For the exercises 3-6, find the exact value.

3) cos(7π12)

Answer

264

4) tan(3π8)

5) tan(sin1(22)+tan13)

Answer

23

6) 2sin(π4)sin(π6)

For the exercises 7-16, find all exact solutions to the equation on [0,2π).

7) cos2xsin2x1=0

Answer

0,π

8) cos2x=cosx

Answer

sin1(14(131)),πsin1(14(131))

9) cos(2x)+sin2x=0

10) 2sin2xsinx=0

Answer

0,π6,5π6,π

11) Rewrite the expression as a product instead of a sum: cos(2x)+cos(8x)

12) Find all solutions of tan(x)3=0.

Answer

π3+kπ

13) Find the solutions of sec2x2secx=15 on the interval [0,2π) algebraically; then graph both sides of the equation to determine the answer.

14) Find sin(2θ), cos(2θ), and tan(2θ) given cotθ=34 and θ is on the interval [π2,π].

Answer

2425,725,247

15) Find sin(θ2), cos(θ2), and tan(θ2) given cosθ=725 and θ is in quadrant IV.

16) Rewrite the expression sin4x with no powers greater than 1.

Answer

18(3+cos(4x)4cos(2x))

For the exercises 17-19, prove the identity.

17) tan3xtanxsec2x=tan(x)

18) sin(3x)cosxsin(2x)=cos2xsinxsin3x

Answer

sin(3x)cosxsin(2x)=sin(x+2x)cosx(2sinxcosx)=sinxcos(2x)+sin(2x)cosx2sinxcos2x=sinx(cos2xsin2x)+2sinxcosxcosx2sinxcos2x=sinxcos2xsin3x+0=cos2xsinxsin3x=cos2xsinxsin3x

19) sin(2x)sinxcos(2x)cosx=secx

20) Plot the points and find a function of the form y=Acos(Bx+C)+D that fits the given data.

x 0 1 2 3 4 5
y -2 2 -2 2 -2 2
Answer

y=2cos(πx+π)

21) The displacement h(t) in centimeters of a mass suspended by a spring is modeled by the function h(t)=14sin(120πt), where t is measured in seconds. Find the amplitude, period, and frequency of this displacement.

22) A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

Answer

81.5,78.7

23) Two frequencies of sound are played on an instrument governed by the equation n(t)=8cos(20πt)cos(1000πt).What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

24) The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

Answer

6+5cos(π6(1x)). From November 23 to February 6.

25) A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring t seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

26) Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water t months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

Answer

D(t)=2cos(π6t)+108+14t, 93.5855 months (or 7.8 years) from now

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This page titled 1.7.R: Trigonometric Identities and Equations (Review) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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