Worksheet: Amplitude and Period of Trigonometric Functions

In this worksheet, we will practice finding the amplitude and the period of sine, cosine, and tangent functions.

Q1:

Determine the amplitude and the period of the shown function.

  • Aamplitude =3, period =1
  • Bamplitude =3, period =2
  • Camplitude =6, period =1
  • Damplitude =6, period =3
  • Eamplitude =3, period =2

Q2:

What is the period of 𝑓(𝑥)=2𝑥+𝜋3sin?

  • A6𝜋
  • B5𝜋
  • C𝜋3
  • D2𝜋
  • E3𝜋

Q3:

What is the period of 𝑓(𝑥)=2𝑥cos?

  • A2𝜋
  • B𝜋3
  • C𝜋2
  • D𝜋4
  • E𝜋

Q4:

What is the period of 𝑓(𝑥)=233𝑥4cos?

  • A8𝜋
  • B2𝜋3
  • C2𝜋
  • D3𝜋4
  • E8𝜋3

Q5:

Find the period of the function 𝑓(𝜃)=114𝜃sin.

  • A2𝜋
  • B11𝜋2
  • C𝜋2
  • D2𝜋11

Q6:

What is the period of the function 𝑓(𝑥)=𝑎(𝑏𝑥𝑐)+𝑘cos?

  • A𝑎
  • B𝑏
  • C𝑏2𝜋
  • D2𝜋𝑏
  • E𝑐

Q7:

What is the period of 𝑓(𝑥)=32𝑥𝜋5tan?

  • A3𝜋
  • B5𝜋2
  • C2𝜋5
  • D𝜋5
  • E6𝜋

Q8:

What is the amplitude of the function 𝑓(𝑥)=𝑎(𝑏(𝑥))+𝑘sin?

  • A𝑎+𝑘
  • B𝑎
  • C2𝜋𝑏
  • D𝑏
  • E

Q9:

Simplify sin(180𝜃).

  • Asin𝜃
  • Bcos𝜃
  • C𝜃cos
  • D𝜃sin

Q10:

What is the amplitude of the function 𝑓(𝑥)=𝑎(𝑏𝑥𝑐)+𝑘cos?

  • A2𝜋𝑏
  • B𝑏
  • C𝑎+𝑘
  • D𝑐
  • E𝑎

Q11:

A function 𝑓(𝑥) has a period 𝑇. Among the following statements, which one is NOT necessarily true?

  • A𝑓(𝑥+𝑇)=𝑓(𝑥)
  • B𝑓(𝑥𝑛𝑇)=𝑓(𝑥)
  • C𝑓(𝑥+𝑛𝑇)=𝑓(𝑥)
  • D𝑓(𝑛𝑇𝑥)=𝑓(𝑥)

Q12:

What is the maximum value of the function 𝑓(𝑥)=𝑎(𝑏(𝑥))+𝑘sin?

  • A𝑎+𝑘
  • B𝑏
  • C𝑎
  • D2𝜋𝑏
  • E

Q13:

Let 𝑓(𝑥)=2𝑥sin. What is the smallest positive value of 𝑃 for which 𝑓(𝑥+𝑃)=𝑓(𝑥) holds?

  • A𝜋
  • B𝜋3
  • C2𝜋
  • D3𝜋2
  • E𝜋2

Q14:

Since sinsin(𝑥+4𝜋)=𝑥 for every number 𝑥, we can conclude that .

  • AThe period of the sine function is a divisor of 4𝜋.
  • BThe period of the sine function is 3𝜋.
  • CThe period of the function is 1.
  • DThe period of the sine function is 4𝜋.
  • ESine is an even function.

Q15:

What is the period of the function 𝑓(𝑥)=343𝜋𝑥5+4𝜋sin?

  • A510
  • B103
  • C𝜋2
  • D𝜋
  • E310

Q16:

What is the amplitude of 𝑓(𝑥)=54𝜋𝑥75+12cos?

  • A14
  • B5
  • C12
  • D𝜋7
  • E54

Q17:

sin𝑥 has period 2𝜋 and tan𝑥 has period 𝜋. What is the period of their sum sintan𝑥+𝑥?

  • A𝜋
  • B3𝜋
  • C32𝜋
  • D𝜋 and 2𝜋
  • E2𝜋

Q18:

Consider the function 𝑓(𝑥)=7𝑥2𝜋cos.

Determine the amplitude.

  • Aamplitude =7
  • Bamplitude =7
  • Camplitude =14
  • Damplitude =72
  • Eamplitude =72

Determine the period.

  • Aperiod =2𝜋
  • Bperiod =𝜋
  • Cperiod =7𝜋
  • Dperiod =𝜋2
  • Eperiod =4𝜋

Q19:

What is the period of the function 𝑓(𝑥)=𝑥sin?

  • A𝜋3
  • B𝜋
  • C𝜋2
  • D3𝜋
  • E2𝜋

Q20:

What is the period of the function 𝑓(𝑥)=𝑥cos?

  • A3𝜋
  • B𝜋2
  • C𝜋3
  • D𝜋
  • E2𝜋

Q21:

What is the amplitude of the function 𝑦=5(4𝑥+3)sin?

Q22:

What is the amplitude of 𝑓(𝑥)=𝑥sin?

Q23:

What is the frequency of the function 𝑓(𝑡)=𝑡cos?

  • A2𝜋
  • B1𝜋
  • C𝜋
  • D12𝜋
  • E3𝜋2

Q24:

A point 𝑅 is traveling on a unit circle at a speed of one complete rotation per second.

If its position represents an angle 𝜃 at a time 𝑡, when will it have the same position again?

  • Aone and a half seconds later
  • Bone-third of a second later
  • Cone second later
  • Dhalf a second later
  • Etwo seconds later

How would you express the angle representing its position at this time?

  • A𝜃+2𝜋
  • B𝜃+𝜋3
  • C𝜃+𝜋2
  • D𝜃+𝜋4
  • E𝜃+𝜋

Deduce what the period of the cosine and sine functions is.

  • A𝜋3
  • B2𝜋
  • C𝜋2
  • D𝜋4
  • E𝜋

Q25:

What is the maximum value of 𝑓(𝑥)=𝑥sin?

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