Trig: Past Paper Style (GDC)
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Solve \(3\sin x=2\cos x+1\) for \(0\leq x\leq2\pi\). Give answers to 3 significant figures.
Rearrange or use GDC graphically
Write as \(3\sin x-2\cos x=1\); use \(R\sin(x-\alpha)=1\) form or GDC
Answer
\(x\approx0.869\) or \(x\approx3.45\)
Question 2
The function \(f(x)=2\sin(3x)+1\) models a wave. Find the period, amplitude, and all zeros in \([0,2\pi]\).
Period and amplitude
Period \(=\dfrac{2\pi}{3}\); Amplitude \(=2\)
Zeros
\(2\sin(3x)+1=0\Rightarrow\sin(3x)=-\tfrac{1}{2}\Rightarrow 3x=\pi+\tfrac{\pi}{6},\;2\pi-\tfrac{\pi}{6},\ldots\)
Answer
Period \(\dfrac{2\pi}{3}\); amplitude 2; zeros at \(x=\dfrac{7\pi}{18},\dfrac{11\pi}{18},\dfrac{19\pi}{18},\dfrac{23\pi}{18},\dfrac{31\pi}{18},\dfrac{35\pi}{18}\)
Question 3
A Ferris wheel of radius 15 m has its centre 17 m above ground. It completes one rotation every 40 seconds. A person boards at the lowest point.
(a)Write a function \(h(t)\) for the height above ground after \(t\) seconds.
(b)Find the times in the first 40 seconds when the person is above 28 m.
(a)
Answer
\(h(t)=17-15\cos\!\left(\dfrac{\pi t}{20}\right)\)
(b)
Solve \(h(t)>28\)
\(-15\cos\!\left(\tfrac{\pi t}{20}\right)>11\Rightarrow\cos\!\left(\tfrac{\pi t}{20}\right)<-\tfrac{11}{15}\Rightarrow\tfrac{\pi t}{20}\in(\arccos(-\tfrac{11}{15}),\;2\pi-\arccos(-\tfrac{11}{15}))\)
Answer
Approx \(t\in(15.2,\;24.8)\) seconds
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