Worksheet: Amplitude and Period of Trigonometric Functions

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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?

  • A 6 𝜋
  • B 5 𝜋
  • C 𝜋 3
  • D 2 𝜋
  • E 3 𝜋

Q3:

What is the period of 𝑓(𝑥)=2𝑥cos?

  • A 2 𝜋
  • B 𝜋 3
  • C 𝜋 2
  • D 𝜋 4
  • E 𝜋

Q4:

What is the period of 𝑓(𝑥)=233𝑥4cos?

  • A 8 𝜋
  • B 2 𝜋 3
  • C 2 𝜋
  • D 3 𝜋 4
  • E 8 𝜋 3

Q5:

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

  • A 2 𝜋
  • B 1 1 𝜋 2
  • C 𝜋 2
  • D 2 𝜋 1 1

Q6:

What is the period of the function 𝑓(𝑥)=𝑎(𝑏𝑥−𝑐)+𝑘cos?

  • A 𝑎
  • B 𝑏
  • C 𝑏 2 𝜋
  • D 2 𝜋 𝑏
  • E 𝑐

Q7:

What is the period of 𝑓(𝑥)=32𝑥−𝜋5tan?

  • A 3 𝜋
  • B 5 𝜋 2
  • C 2 𝜋 5
  • D 𝜋 5
  • E 6 𝜋

Q8:

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

  • A 𝑎 + 𝑘
  • B 𝑎
  • C 2 𝜋 𝑏
  • D 𝑏
  • E ℎ

Q9:

Simplify sin(180−𝜃)∘.

  • A s i n 𝜃
  • B c o s 𝜃
  • C − 𝜃 c o s
  • D − 𝜃 s i n

Q10:

What is the amplitude of the function 𝑓(𝑥)=𝑎(𝑏𝑥−𝑐)+𝑘cos?

  • A 2 𝜋 𝑏
  • B 𝑏
  • C 𝑎 + 𝑘
  • D 𝑐
  • E 𝑎

Q11:

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

  • A 𝑓 ( 𝑥 + 𝑇 ) = 𝑓 ( 𝑥 )
  • B 𝑓 ( 𝑥 − 𝑛 𝑇 ) = 𝑓 ( 𝑥 )
  • C 𝑓 ( 𝑥 + 𝑛 𝑇 ) = 𝑓 ( 𝑥 )
  • D 𝑓 ( 𝑛 𝑇 − 𝑥 ) = 𝑓 ( 𝑥 )

Q12:

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

  • A 𝑎 + 𝑘
  • B 𝑏
  • C 𝑎
  • D 2 𝜋 𝑏
  • E ℎ

Q13:

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

  • A 𝜋
  • B 𝜋 3
  • C 2 𝜋
  • D 3 𝜋 2
  • E 𝜋 2

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?

  • A 5 1 0
  • B 1 0 3
  • C 𝜋 2
  • D 𝜋
  • E 3 1 0

Q16:

What is the amplitude of 𝑓(𝑥)=54𝜋𝑥7−5+12cos?

  • A14
  • B5
  • C12
  • D 𝜋 7
  • E 5 4

Q17:

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

  • A 𝜋
  • B 3 𝜋
  • C 3 2 𝜋
  • D 𝜋 and 2𝜋
  • E 2 𝜋

Q18:

Consider the function 𝑓(𝑥)=−7𝑥2−𝜋cos.

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?

  • A 𝜋 3
  • B 𝜋
  • C 𝜋 2
  • D 3 𝜋
  • E 2 𝜋

Q20:

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

  • A 3 𝜋
  • B 𝜋 2
  • C 𝜋 3
  • D 𝜋
  • E 2 𝜋

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?

  • A 2 𝜋
  • B 1 𝜋
  • C 𝜋
  • D 1 2 𝜋
  • E 3 𝜋 2

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?

  • A 𝜃 + 2 𝜋
  • B 𝜃 + 𝜋 3
  • C 𝜃 + 𝜋 2
  • D 𝜃 + 𝜋 4
  • E 𝜃 + 𝜋

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

  • A 𝜋 3
  • B 2 𝜋
  • C 𝜋 2
  • D 𝜋 4
  • E 𝜋

Q25:

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

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