Solving Trig Equations (Non-GDC)
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Solve for \(0\leq heta\leq360°\), giving exact answers:
(a)\(\sin heta=\dfrac{\sqrt{3}}{2}\)
(b)\(\cos heta=-\dfrac{1}{\sqrt{2}}\)
(c)\( an heta=\sqrt{3}\)
(a)
Answer
\( heta=60°\) or \(120°\)
(b)
Answer
\( heta=135°\) or \(225°\)
(c)
Answer
\( heta=60°\) or \(240°\)
Question 2
Solve for \(0\leq x\leq2\pi\):
(a)\(2\sin x-\sqrt{3}=0\)
(b)\(2\cos^2 x-\cos x=0\)
(c)\( an^2 x=3\)
(a)
Rearrange
\(\sin x= frac{\sqrt3}{2}\)
Answer
\(x=\dfrac{\pi}{3}\) or \(\dfrac{2\pi}{3}\)
(b)
Factorise
\(\cos x(2\cos x-1)=0\Rightarrow\cos x=0\) or \(\cos x= frac{1}{2}\)
Answer
\(x=\dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{\pi}{3},\dfrac{5\pi}{3}\)
(c)
Square root
\( an x=\pm\sqrt{3}\)
Answer
\(x=\dfrac{\pi}{3},\dfrac{2\pi}{3},\dfrac{4\pi}{3},\dfrac{5\pi}{3}\)
Question 3
Solve for \(0\leq x\leq2\pi\): \(2\sin^2 x+3\sin x-2=0\)
Factorise
\((2\sin x-1)(\sin x+2)=0\Rightarrow\sin x= frac{1}{2}\) (reject \(\sin x=-2\))
Answer
\(x=\dfrac{\pi}{6}\) or \(\dfrac{5\pi}{6}\)
Question 4
Solve for \(-\pi\leq heta\leq\pi\): \(\sin 2 heta=\cos heta\)
Substitute \(\sin2 heta=2\sin heta\cos heta\)
\(2\sin heta\cos heta-\cos heta=0\Rightarrow\cos heta(2\sin heta-1)=0\)
Solve
\(\cos heta=0\Rightarrow heta=\pm frac{\pi}{2}\); \(\sin heta= frac{1}{2}\Rightarrow heta= frac{\pi}{6},\; frac{5\pi}{6}\)
Answer
\( heta=-\dfrac{\pi}{2},\;\dfrac{\pi}{6},\;\dfrac{\pi}{2},\;\dfrac{5\pi}{6}\)
Question 5
Find the general solution of \(\cos 3x = \dfrac{1}{2}\).
Principal value
\(3x=\dfrac{\pi}{3}+2k\pi\) or \(3x=-\dfrac{\pi}{3}+2k\pi\), \(k\in\mathbb{Z}\)
Answer
\(x=\dfrac{\pi}{9}+\dfrac{2k\pi}{3}\) or \(x=-\dfrac{\pi}{9}+\dfrac{2k\pi}{3}\), \(k\in\mathbb{Z}\)
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