Circular functions revision

Circular functions revision

by Sophie Bark

​The unit circle

​The unit circle is the starting point for your revision...

​The unit circle

​Finding angles in different quadrants

​Example - 

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​Find the value of

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a)

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​b)

​

​c)

​Finding angles in different quadrants

​Example - 

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​Find the value of...

​

​

​b)

​

​c)

Multiple Choice

Which other angle gives the same value as  $$\sin\frac{\pi}{4}$$ ?

• $$\frac{3\pi}{4}$$  

• $$\frac{5\pi}{4}$$  

• $$\frac{7\pi}{4}$$  

• $$\frac{11\pi}{4}$$  

Multiple Choice

If $$\sin\theta=0.68$$  , find the value of $$\sin\left(\pi+\theta\right)$$  

• 0.68

• -0.68

• 0.32

• -0.32

Multiple Choice

If $$\tan x=\tan\left(\frac{\pi}{6}\right)\ ,\ \pi\le x\le2\pi$$  , find $$x$$  

• $$x=\frac{\pi}{6}$$  

• $$x=\frac{5\pi}{6}$$  

• $$x=\frac{7\pi}{6}$$  

• $$x=\frac{11\pi}{6}$$  

Multiple Choice

What is the length of the arc on the unit circle between $$0\degree$$  and $$90\degree$$  ?

• $$\pi$$  cm

• $$\ 2\pi$$  cm

• $$\frac{\pi}{2}$$  cm

• $$\frac{\pi}{4}$$  cm

Multiple Choice

What is the length of an arc formed by an angle of  $$60\degree$$  in a circle with radius 8cm?

• $$\frac{8\pi}{3}cm$$  

• $$\frac{32\pi}{3}cm$$  

• $$\frac{\pi}{3}cm$$  

• $$\frac{\pi}{6}cm$$  

​Solving Equations to find solutions in a specified domain

​The test is going to be CAS active, but you may need to show working out for some questions.

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​We will practise the CAS skills first!!

​Solve the equation using CAS

​Solve the equation using CAS - check your answer

Fill in the Blank

These values of x would be the ________ of the graph of  $$y=2\sin\left(2x\right)-1$$   $$0\le x\le2\pi$$  

Open Ended

What is the amplitude and period of the graph $$y=2\sin\left(2x\right)-1$$  

​Amplitude and period

​The amplitude is the distance from the mean (middle) of the graph to the maximum of minimum.

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​The period is how long it takes for the graph to make one complete cycle.

Multiple Choice

What is the range of  $$y=2\sin\left(2x\right)-1$$  

• $$\left[-2,2\right]$$  

• $$\left[-1,1\right]$$  

• $$\left[-1,3\right]$$  

• $$\left[-3,1\right]$$  

Multiple Choice

Which is the correct sketch of $$y=2\sin\left(2x\right)-1$$  ?

​Now for some CAS practise

Multiple Choice

Solve the equation finding all solutions

$$2\cos\left(3x\right)-\sqrt{3}=0\ ,\ x\in\left[0,2\pi\right]$$  

• $$x=\frac{\pi}{18},\frac{11\pi}{18},\frac{13\pi}{18},\ \frac{23\pi}{18},\ \frac{25\pi}{18},\frac{35\pi}{18}$$  

• $$x=\frac{\pi}{18},\ \ \frac{11\pi}{18}$$  

• $$x=\ \frac{\pi}{18},\ \frac{11\pi}{18},\ \frac{13\pi}{18},\ \frac{23\pi}{18},\ \frac{25\pi}{18},\ \ \frac{35\pi}{18},\frac{37\pi}{18},\frac{47\pi}{18}$$  

• $$x=\frac{\pi}{18},\ \ \frac{11\pi}{18},\ \ \frac{23\pi}{18},\ \ \frac{35\pi}{18}$$  

Multiple Choice

What is the smallest value of $$f\left(x\right)=5-\sin x$$  

If using CAS, you must include a domain (hint: one complete cycle will contain 1 minimum & 1 maximum)

• $$-1$$  

• $$4$$  

• $$-5$$  

• $$-4$$  

​Sequences of Transformations

​Sequences of Transformations

Open Ended

State the sequence of transformations that takes $$y=\tan x$$  to $$y=-\tan\left(3x\right)-5$$  

​Dilation by factor 1/3 from y-axis

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​Reflection in the x-axis

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​Translation 5 units in the negative direction of y-axis

Open Ended

State the sequence of transformations that takes  $$y=\cos x$$  to  $$y=3\cos\left(\frac{1}{2}\left(x-\frac{\pi}{4}\right)\right)+7$$  

​1. Translation pi/4 units in the positive direction of the x-axis (right)

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​2. Dilation by factor 2 from the y-axis

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​3. Dilation by factor 3 from the x-axis

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​4. Translation 7 units in the positive direction of y-axis (up)

Poll

How confident do you feel with circular functions?

• Very confident with all aspects

• Confident with the concepts, but I need to practise

• I understand most things, but I need to practise. There are still one or two things I don't understand.

• I understand about half, and there is half that I don't understand.

• I don't understand most of it

Open Ended

What do you need the most help with? Please try and be specific here, e.g.

I can solve, but I struggle to find all solutions in the domain..

I find it hard to sketch when there is a vertical translation

I don't understand the difference between the amplitude and the range

etc