Exercises for Question Type 4: Converting between sinusoidal model representations

Question 1

Skill question

This question requires simplifying a trigonometric function by reducing the angle modulo $2 π$ , converting a sine function to a cosine function using trigonometric identities, and determining the phase shift within a given interval.

Simplify $y = sin (6 x + \frac{17 π}{3})$ and express it in the form $cos (6 (x - i))$ . Find a value of $i$ in the interval $[0, 2 π)$ .

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Question 2

Skill question

Derive the general formula for writing $sin (ω x + α)$ in the form $cos (ω (x - c))$ , expressing $c$ in terms of $α$ and $ω$ .

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Question 3

Skill question

Express $y = 4 cos (5 x - \frac{7 π}{3})$ in the form $y = 4 sin (5 (x + q))$ , where $q$ is the smallest positive constant.

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Question 4

Skill question

Rewrite $y = 5 cos (2 θ + \frac{π}{3})$ in the form $A sin (2 (θ + e))$ and find the values of $A$ and $e$ .

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Question 5

Skill question

Express $y = 3 sin (4 x - \frac{7 π}{3})$ in the form $y = 3 cos (4 (x - c))$ and find the smallest positive value of $c$ .

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Question 6

Skill question

Express $y = 4 sin (3 x - \frac{π}{4})$ in the form $4 cos (3 (x - d))$ and find the value of $d$ .

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Question 7

Skill question

Express $y = 7 cos (2 ϕ + \frac{3 π}{4})$ in the form $y = 7 sin (2 (ϕ + h))$ and find the value of $h$ .

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Question 8

Skill question

Express $y = - 5 sin (3 t - \frac{5 π}{6})$ in the form $5 cos (3 (t - g))$ and determine the value of $g$ .

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Question 9

Skill question

The question asks for the conversion of a sine function with a phase shift and horizontal compression into a cosine function of a specific form, identifying a parameter within a given range.

Convert $y = sin (5 x + 9 π)$ into the form $cos (5 (x - c))$ and give a value for $c$ in the interval $0 \leq c < 2 π$ .

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Question 10

Skill question

Write $y = sin (2 x + \frac{π}{6})$ in the form $cos (2 (x - c))$ and determine the value of $c$ .

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Question 11

Skill question

Rewrite $y = 3 cos (2 x + \frac{5 π}{6})$ in the form $3 sin (2 (x - c))$ and determine the value of $c$ for $c \in [0, π)$ .

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Question 12

Skill question

Write $y = 2 cos (4 t - π)$ in the form $2 sin (4 (t - f))$ and find $f$ .

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IB Mathematics Applications & Interpretation Question Type 4: Converting Between Sinusoidal Model Representations Exercises