PreCalc 2 · Trigonometric Functions
Sine and cosine models go up and down in a steady cycle. Read max, min, midline, and period from the story.
What the pieces mean
Midline is the middle value. Amplitude is the distance from the middle to the top.
Period is how long one full cycle takes (one Ferris-wheel turn, one tide cycle).
Example
y = 5 sin(t) + 12 has midline 12, max 17, min 7.
y = 5 sin(t) + 12. What is the maximum value?
Max = midline + amplitude = 12 + 5 = 17.
Answer: 17
y = 4 sin(t) + 10. What is the minimum value?
Min = 10 − 4 = 6.
Answer: 6
y = 3 cos(t) + 8. What is the midline?
The +8 is the vertical shift, so midline is 8.
Answer: 8
y = 6 sin(t) + 20. What is the amplitude?
The 6 in front of sine.
Answer: 6
y = −2 sin(t) + 9. What is the maximum value?
Amplitude is still 2. Max = 9 + 2 = 11.
Answer: 11
y = 7 cos(t) + 1. What is the minimum value?
Min = 1 − 7 = −6.
Answer: −6
A Ferris wheel height is h = 30 sin(2πt / 40) + 35, with t in seconds. How long is one full turn?
Period is 40 seconds (the number under 2πt).
Answer: 40 seconds
That same wheel: what is the highest point?
Midline 35, amplitude 30. Max = 65.
Answer: 65
That same wheel: what is the lowest point?
Min = 35 − 30 = 5.
Answer: 5
A tide is y = 4 cos(2πt / 12) + 6, t in hours. How many hours between high tides?
One full cycle is 12 hours.
Answer: 12 hours
In that tide model, what is the midline water height?
The +6.
Answer: 6
Temperature y = 11 sin(t) + 64. What is the range of temperatures?
Min 64 − 11 = 53. Max 64 + 11 = 75.
Answer: 53 to 75
A person’s blood pressure is modeled by y = 20 sin(2πt) + 100, t in seconds. What do 20 and 100 mean?
100 is the middle pressure. 20 is how far it goes above and below that middle.
Answer: midline 100, amplitude 20
A spring’s height is y = 8 cos(πt) + 3. Find the period.
B = π, period = 2π/π = 2 seconds.
Answer: 2 seconds
Write a sine model with midline 15, amplitude 4, and period 8. Use y = A sin(Bt) + D.
A = 4, D = 15, B = 2π/8 = π/4.
y = 4 sin(πt/4) + 15.
Answer: y = 4 sin(πt/4) + 15
Daylight hours are y = 3 sin((2π/365)t) + 12. What is the most daylight in a day?
Max = 12 + 3 = 15 hours.
Answer: 15 hours
A wheel’s height is y = −25 cos(2πt/10) + 30. Why might the cosine be negative?
A negative cosine starts at the bottom (or reflected). Here it can model starting at the low point.
Answer: starts at a low point (reflection)
Write a cosine model with max 18, min 2, and period 6.
Midline = (18+2)/2 = 10. Amplitude = 8. B = 2π/6 = π/3.
y = 8 cos(πt/3) + 10.
Answer: y = 8 cos(πt/3) + 10
The angle of elevation of the sun is 36°. A tree casts a 50-foot shadow. To the nearest foot, how tall is the tree?
tan 36° = height / 50.
height = 50 tan 36° ≈ 36.327.
To the nearest foot: 36 ft.
Answer: 36 ft
A tower is 75 feet tall and casts a 100-foot shadow. Find the angle of elevation of the sun to the nearest degree.
tan θ = 75/100 = 0.75.
θ = tan⁻¹(0.75) ≈ 36.87°.
To the nearest degree: 37°.
Answer: 37°
A balloon rises vertically. An observer 80 feet from the point on the ground directly under the balloon watches the angle of elevation change from 18° to 29°. How far, to the nearest tenth of a foot, does the balloon rise?
First height = 80 tan 18°. Second height = 80 tan 29°.
Rise = 80(tan 29° − tan 18°) ≈ 18.351.
To the nearest tenth: 18.4 ft.
Answer: 18.4 ft
Write the compass bearing: a ray that is 35° west of south.
Start at south, then 35° toward west.
S 35° W.
Answer: S 35° W
Write the compass bearing: a ray that is 70° east of north.
Start at north, then 70° toward east.
N 70° E.
Answer: N 70° E
Write the compass bearing: a ray that is 15° east of south.
Start at south, then 15° toward east.
S 15° E.
Answer: S 15° E
Need a little more help with this skill?
Book a Session with Amy