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Let \((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\).
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a. \(\tan\,\theta = 1\ \ \text{or}\ \ \tan\,\theta = \pm \sqrt {3}\)
b. \(\theta = 45^{\circ},60^{\circ},120^{\circ}\)
a. \((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\)
\(\Rightarrow \tan\,\theta = 1\)
| \(\Rightarrow \sin\,\theta-\sqrt{3}\cos\,\theta\) | \(=0\) |
| \(\sin\,\theta\) | \(=\sqrt{3}\cos\,\theta\) |
| \(\tan\,\theta\) | \(=\sqrt{3}\) |
| \(\Rightarrow\sin\,\theta + \sqrt{3}\cos\,\theta\) | \(=0\) |
| \(\sin\,\theta\) | \(=-\sqrt{3}\cos\,\theta\) |
| \(\tan\,\theta\) | \(=-\sqrt{3}\) |
\(\therefore \tan\,\theta = 1\ \ \text{or}\ \ \tan\,\theta = \pm \sqrt{3}\)
b. \((\tan\,\theta-1)(\sin^2\theta-3\cos^2\theta) = 0\)
\(\text{Using part a:}\)
\((\tan\,\theta-1) (\sin\,\theta-\sqrt{3}\cos\,\theta)(\sin\,\theta + \sqrt{3}\cos\,\theta) = 0\)
| \(\Rightarrow \tan\,\theta\) | \(= 1\) | \(\ \ \ \ \ \ \ \ \ \) | \(\tan\,\theta\) | \(= \pm \sqrt{3}\) |
| \(\theta\) | \(= 45^{\circ}\) | \(\theta\) | \(= 60^{\circ}, 120^{\circ}\) |
\(\therefore \theta = 45^{\circ},60^{\circ},120^{\circ}\ \ \ \ (0^{\circ} \leq \theta \leq 180^{\circ})\)
Solve \(2\sin\,\theta\,\cos\,\theta=\sin\,\theta\) for \(0^{\circ} \leq \theta \leq 360^{\circ}\). (3 marks)
\(\theta=0^{\circ}, 60^{\circ}, 180^{\circ}, 300^{\circ}, 360^{\circ}\)
♦ Mean mark 49%.
| \(2\sin\,\theta\,\cos\,\theta\) | \(=\sin\,\theta\) |
| \(2\sin\,\theta\,\cos\,\theta-\sin\,\theta\) | \(= 0\) |
| \(\sin\,\theta(2\cos\,\theta-1)\) | \(=0\) |
\(\sin\,\theta=0:\)
\(\Rightarrow \theta=0^{\circ}, 180^{\circ}, 360^{\circ}\)
\(\cos\,\theta=\dfrac{1}{2}:\)
\(\text{Reference angle}\ =60^{\circ}\)
\(\text{cos is positive in 1st/4th quadrants.}\)
\(\Rightarrow \theta=60^{\circ}, (360-60)^{\circ} = 60^{\circ}, 300^{\circ}\)
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a. \(\text{Prove}\ \cos\,\theta\,\tan\,\theta = \sin\,\theta\)
| \(\text{LHS}\) | \(= \cos\,\theta\,\tan\,\theta\) |
| \(=\cos\,\theta \times \dfrac{\sin\,\theta}{\cos\,\theta}\) | |
| \(= \sin\,\theta\) | |
| \(=\ \text{RHS}\) |
| b. | \(8\sin\,\theta\,\cos\,\theta\,\tan\,\theta\) | \(= \dfrac{1}{\sin\,\theta}\) |
| \(8\sin^{2}\theta\) | \(= \dfrac{1}{\sin\,\theta}\) | |
| \(8\sin^{3}\theta\) | \(= 1\) | |
| \(\sin^{3}\theta\) | \(=\dfrac{1}{8}\) | |
| \(\sin\,\theta\) | \(=\dfrac{1}{2}\) |
\(\text{Reference angle:}\ \theta=30^{\circ}\)
\(\text{sin is positive in 1st/2nd quadrants.}\)
\(\therefore \ \theta=30^{\circ}, (180-30)^{\circ}=30^{\circ}, 150^{\circ}\)
Solve the equation \(\sqrt{3}\sin\,\theta = \cos\,\theta\) for \(-180^{\circ} \leq \theta \leq 180^{\circ}\). (2 marks)
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\(\theta = 30^{\circ}\ \text{or}\ -150^{\circ}\)
\(\text{Divide both sides by}\ \cos\,\theta:\)
| \(\sqrt{3}\sin\,\theta\) | \(=\cos\,\theta\) |
| \(\sqrt{3}\tan\,\theta\) | \(= 1\) |
| \(\tan\,\theta\) | \(= \dfrac {1}{\sqrt{3}}\) |
\(\text{Reference angle:}\ \theta=30^{\circ}\)
\(\therefore\ \theta = 30^{\circ}\ \text{or}\ -150^{\circ}, \ \text{for }-180^{\circ} \leq \theta \leq 180^{\circ}\)
Find all solutions of the equation \(2 \cos\,\theta = \dfrac{\sqrt{3}}{\tan\,\theta}\), for \(0^{\circ} \leq \theta \leq 360^{\circ}\) (3 marks)
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\(\theta = 60^{\circ}, 90^{\circ}, 120^{\circ}, 270^{\circ}\)
\(2 \cos\,\theta = \dfrac{\sqrt{3}}{\tan\,\theta}\)
| \(2 \cos\,\theta-\dfrac{\sqrt{3}\cos\,\theta}{\sin\,\theta}\) | \(= 0\) | |
| \(\cos\,\theta\left(2-\dfrac{\sqrt{3}}{\sin\,\theta}\right) \) | \(=0\) |
\(\text{If}\ \ \cos\,\theta=0:\)
\(\Rightarrow\ \ \theta=90^{\circ}, 270^{\circ}\)
\(\text{If}\ \ 2-\dfrac{\sqrt{3}}{\sin\,\theta} = 0\ \Rightarrow\ \ \sin\,\theta = \dfrac{\sqrt{3}}{2}:\)
\(\Rightarrow\ \ \theta = 60^{\circ}, 120^{\circ}\)
Solve \(\sin x-\cos x=0 \quad-180^{\circ} \leqslant x \leqslant 180^{\circ}\) (2 marks)
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\(x=45^{\circ}, -135^{\circ}\)
| \(\sin x-\cos x\) | \(=0\) | |
| \(\dfrac{\sin x}{\cos x}-\dfrac{\cos x}{\cos x}\) | \(=0\) | |
| \(\tan x-1\) | \(=0\) | |
| \(\tan x\) | \(=1\) | |
| \(x\) | \(=\tan ^{-1}(1)\) |
\(\therefore x=45^{\circ}, -135^{\circ}\)
Express \(\dfrac{3}{\sin(180+\theta)}+5 \cos (90-\theta)\) as a single fraction in terms of \(\sin \theta\), given all angles are measured in degrees. (3 marks) --- 8 WORK AREA LINES (style=lined) --- \(\dfrac{-3+5 \sin ^2 \theta}{\sin\,\theta}\) \(\dfrac{3}{\sin(180+\theta)}+5 \cos (90-\theta)\) \(=-\dfrac{3}{\sin\,\theta}+5 \sin\,\theta\) \(=-\dfrac{3}{\sin\,\theta}+\dfrac{5\sin^{2}\theta}{\sin\,\theta}\) \(=\dfrac{-3+5 \sin ^{2}\theta}{\sin\,\theta}\)
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How many solutions of the equation `(sin theta-1)(tan theta + 2) = 0` lie between `0°` and `360°`?
`B`
`text(When)\ (sin theta-1)(tan theta + 2) = 0`
`(sin theta-1) = 0\ \ text(or)\ \ tan theta + 2 = 0`
`text(If)\ \ sin theta-1= 0:`
`sin theta= 1\ \ =>\ \ theta= 90°,\ \ \ 0° < theta < 360°`
`text(If)\ \ tan theta + 2= 0:`
`tan theta= -2`
`text{Since}\ tan\ 90°\ text{is undefined, there are only 2 solutions when}`
`tan theta = -2\ \text{(which occurs in the 1st and 4th quadrants).}`
`:.\ 2\ text(solutions)`
`=> B`
Determine the number of values of \(\theta\) in the range \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\) that satisfy the equation
\((\tan \theta-\sqrt{3})(\cos^{2}\theta-1)=0 \)
\(C\)
\(\tan \theta = \sqrt{3}\ \rightarrow \text{2 solutions} \)
| \(\cos^{2}\theta\) | \(=1\) | |
| \(\cos\theta\) | \(= \pm 1\) | |
| \(\theta\) | \(=0^{\circ}, 180^{\circ}, 360^{\circ}\ \rightarrow \text{3 solutions} \) |
\(\Rightarrow C\)