Grade 11 trigonometry: 17 questions with diagrams where needed and searchable worked solutions. Official source links are shown only where matched.
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3 questions
Simplify: \(\sin(180°-\theta)\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| For an acute reference angle, use the indicated quadrant. The resulting reduction identity holds for every real angle. | 1 |
| \(\boxed{\sin(180°-\theta)=\sin\theta}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Simplify: \(\cos(360°-\theta)\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| For an acute reference angle, use the indicated quadrant. The resulting reduction identity holds for every real angle. | 1 |
| \(\boxed{\cos(360°-\theta)=\cos\theta}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
For \(y=2\sin x\), state the amplitude and the range.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| Amplitude \(=|2|=\boxed{2}\) | 1 |
| Range \(=\boxed{[-2;2]}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
10 questions
If \(\sin\theta=0{,}6\), determine \(\cos^2\theta\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin^2\theta+\cos^2\theta=1\Rightarrow\cos^2\theta=1-0{,}36=\boxed{0{,}64}\) | 1 |
| Total | 1 |
Suggested practice allocation only. These marks are not copied from an official memo.
Write down the general solution for \(\tan\theta=1\) (\(n\in\mathbb{Z}\)).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| tan has period \(180°\) and reference angle \(45°\) | 1 |
| \(\boxed{\theta=45°+180°n}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Solve for \(\theta\) (general solution, \(n\in\mathbb{Z}\)): \(2\cos\theta=1\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\cos\theta=\dfrac12\), reference angle \(60°\) | 1 |
| \(\boxed{\theta=\pm60°+360°n}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
In \(\triangle ABC\), \(\hat A=40°\), \(\hat B=65°\) and \(BC=8\)cm. Determine \(AC\), correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\frac{AC}{\sin65^\circ}=\frac{8}{\sin40^\circ}\) | 1 |
| \(AC=\frac{8\sin65^\circ}{\sin40^\circ}\approx11.28\text{ cm}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
In \(\triangle PQR\), \(PQ=6\), \(QR=9\) and \(\hat Q=50°\). Determine \(PR\), correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(PR^2=6^2+9^2-2(6)(9)\cos50^\circ\) | 1 |
| \(PR=\sqrt{117-108\cos50^\circ}\approx6.90\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Solve for \(\theta\in[0°;360°]\): \(\sin\theta=-0{,}5\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| Reference angle \(30°\); sine is negative in Q3 and Q4 | 1 |
| \(\theta=180°+30°=210°\) or \(\theta=360°-30°=330°\) | 1 |
| \(\boxed{\theta\in\{210°;330°\}}\) | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
In \(\triangle DEF\), \(DE=10\), \(EF=14\) and \(DF=17\). Determine \(\hat E\), correct to 1 decimal place.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\cos\hat E=\dfrac{DE^2+EF^2-DF^2}{2(DE)(EF)}=\dfrac{100+196-289}{2(10)(14)}=\dfrac{7}{280}=0{,}025\) | 1 |
| \(\hat E=\cos^{-1}(0{,}025)=\boxed{88{,}6°}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
In \(\triangle XYZ\), \(XY=9\)cm, \(XZ=11\)cm and \(\hat X=55°\). Determine the area of \(\triangle XYZ\), correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\text{Area}=\frac12(9)(11)\sin55^\circ\) | 1 |
| \(\text{Area}\approx40.55\text{ cm}^2\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Two ships leave a harbour \(H\) at the same time along different courses, with \(\angle AHB=65°\). After 1 hour, \(HA=18\)km and \(HB=24\)km. Determine the distance \(AB\) between the ships, correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(AB^2=18^2+24^2-2(18)(24)\cos65^\circ\) | 1 |
| \(AB=\sqrt{900-864\cos65^\circ}\approx23.13\text{ km}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
The graph of \(y=a\cos x+q\) has a maximum value of \(5\) and a minimum value of \(-1\). Determine the values of \(a\) and \(q\) (assume \(a>0\)).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| Maximum: \(a+q=5\). Minimum: \(-a+q=-1\) | 1 |
| Adding the two equations: \(2q=4\Rightarrow q=2\) | 1 |
| Substituting back: \(a=5-2=\boxed{3}\), so \(a=3\), \(q=2\) | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
4 questions
If \(\cos27°=k\), determine \(\tan153°\) in terms of \(k\).
North West November 2023, Q6.2.2. Question, answer and marking points checked against the supplied marking guideline.
| Working and reasons | Marks |
|---|---|
| \(153°=180°-27°\), so \(\tan153°=\tan(180°-27°)=-\tan27°\) | 1 |
| \(\cos27°=k\Rightarrow\sin27°=\sqrt{1-k^2}\Rightarrow\tan27°=\dfrac{\sqrt{1-k^2}}{k}\) | Support |
| \(\boxed{\tan153°=-\dfrac{\sqrt{1-k^2}}{k}}\) | 1 |
| Total | 2 |
If \(\sin33°=m\), determine, without using a calculator: (a) \(\tan33°\), (b) \(\cos777°\), (c) \(\sin(-237°)\), all in terms of \(m\).
KZN November 2023, Q5.1. Checked against marking-guideline page 7. The square-root step earns the first mark in part (a).
| Working and reasons | Marks |
|---|---|
| \(\sin33°=m\Rightarrow\cos33°=\sqrt{1-m^2}\) | 1 |
| (a) \(\tan33°=\boxed{\dfrac{m}{\sqrt{1-m^2}}}\) | 1 |
| (b) \(777°-720°=57°\), and \(57°=90°-33°\), so \(\cos777°=\cos57°=\cos(90°-33°)=\sin33°=\boxed{m}\) | 3 |
| (c) \(\sin(-237°)=-\sin237°=-\sin(180°+57°)=-(-\sin57°)=\sin57°=\sin(90°-33°)=\cos33°=\boxed{\sqrt{1-m^2}}\) | 3 |
| Total | 8 |
Solve for \(\theta\in[-180°;180°]\): \(2\sin^2\theta-\sin\theta-1=0\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| Let \(s=\sin\theta\): \(2s^2-s-1=0\Rightarrow(2s+1)(s-1)=0\Rightarrow s=-\dfrac12\) or \(s=1\) | 1 |
| \(\sin\theta=1\Rightarrow\theta=90°\) | 1 |
| \(\sin\theta=-\dfrac12\): reference \(30°\), sine negative in Q3/Q4 gives \(210°,330°\); converted into \([-180°;180°]\) these are \(-150°\) and \(-30°\) | 1 |
| \(\boxed{\theta\in\{-150°;-30°;90°\}}\) | 1 |
| Total | 4 |
Suggested practice allocation only. These marks are not copied from an official memo.
In the diagram, \(KLMN\) is a cyclic quadrilateral with \(\angle KLN=30{,}5°\), \(\angle LMN=50°\), \(MN=6\)cm and \(LM=9\)cm. (a) Calculate the length of \(LN\). (b) Show that \(KN=4{,}57\)cm. (c) Calculate the area of \(\triangle KLN\).
North West November 2023, Q8. Question, answer and marking points checked against the supplied marking guideline.
| Working and reasons | Marks |
|---|---|
| (a) Cosine rule: \(LN=\sqrt{9^2+6^2-2(9)(6)\cos50^\circ}\approx\boxed{6{,}90\text{ cm}}\). Keep the unrounded value. | 3 |
| (b) Opposite angles of cyclic \(KLMN\) are supplementary: \(\angle LKN=130^\circ\). By the sine rule, \(KN=\dfrac{LN\sin30{,}5^\circ}{\sin130^\circ}\approx\boxed{4{,}57\text{ cm}}\). | 3 |
| (c) \(\angle KNL=180^\circ-130^\circ-30{,}5^\circ=19{,}5^\circ\). Using unrounded lengths, \(\text{Area}=\dfrac12(KN)(LN)\sin19{,}5^\circ\approx\boxed{5{,}26\text{ cm}^2}\). | 3 |
| Total | 9 |