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 \(R\sin(x-lpha)=1\) form or GDC
Answer
\(xpprox0.876\) or \(xpprox2.76\)
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
\(2\sin(3x)+1=0\Rightarrow\sin(3x)=- frac{1}{2}\Rightarrow 3x=\pi+ frac{\pi}{6},\;2\pi- frac{\pi}{6},\ldots\)
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
\(h(t)=17-15\cos\!\left(\dfrac{\pi t}{20}
ight)\)
(b)
Solve \(h(t)>28\)
\(-15\cos\!\left( frac{\pi t}{20}
ight)>11\Rightarrow\cos\!\left( frac{\pi t}{20}
ight)<- frac{11}{15}\Rightarrow frac{\pi t}{20}\in(rccos(- frac{11}{15}),\;2\pi-rccos(- frac{11}{15}))\)
Answer
Approx \(t\in(14.8,\;25.2)\) seconds
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