Grade 10 trigonometry: 17 questions with diagrams where needed and searchable worked solutions. Official source links are shown only where matched.
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2 questions
Write down the exact value of \(\tan 45°\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\tan 45°=1\) | 1 |
| Total | 1 |
Suggested practice allocation only. These marks are not copied from an official memo.
If \(\cos\theta=\dfrac{3}{5}\), determine \(\sec\theta\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sec\theta=\dfrac{1}{\cos\theta}=\boxed{\dfrac{5}{3}}\) | 1 |
| Total | 1 |
Suggested practice allocation only. These marks are not copied from an official memo.
13 questions
In a right-angled triangle, the side opposite \(\theta\) is \(6\) and the hypotenuse is \(10\). Determine \(\sin\theta\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin\theta=\dfrac{O}{H}=\dfrac{6}{10}=\boxed{\dfrac{3}{5}}\) | 1 |
| Total | 1 |
Suggested practice allocation only. These marks are not copied from an official memo.
Evaluate, without a calculator: \(\sin 60°\times\cos 30°\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin60°=\dfrac{\sqrt3}{2}\), \(\cos30°=\dfrac{\sqrt3}{2}\) | 1 |
| \(\dfrac{\sqrt3}{2}\times\dfrac{\sqrt3}{2}=\dfrac{3}{4}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
In the diagram, \(\triangle ABC\) is right-angled at \(B\), with \(AB=12\)cm and \(BC=5\)cm. Determine, correct to 2 decimal places where needed: (a) the length of \(AC\), (b) \(\sin\hat A\), (c) the size of \(\hat A\).
WC Overberg November 2025, Q3.1. Question, answer and marking points checked against the supplied marking guideline.
| Working and reasons | Marks |
|---|---|
| (a) Pythagoras: \(AC^2=12^2+5^2=169\Rightarrow AC=\boxed{13\text{ cm}}\) | 2 |
| (b) \(BC\) is opposite \(\hat A\): \(\sin\hat A=\dfrac{5}{13}\) | 2 |
| (c) \(\hat A=\sin^{-1}\left(\dfrac{5}{13}\right)=\boxed{22{,}62°}\) | 2 |
| Total | 6 |
In right-angled \(\triangle DEF\), the side opposite \(\hat D\) is \(8\) and the hypotenuse is \(17\). Determine \(\hat D\), correct to 1 decimal place.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin D=\dfrac{8}{17}\). Keep this fraction in your calculator. | 1 |
| \(\hat D=\sin^{-1}\left(\dfrac{8}{17}\right)\approx\boxed{28{,}1^\circ}\). | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Simplify, without a calculator: \(\dfrac{2\sin45°\cos45°}{\tan60°}\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin45°=\cos45°=\dfrac{\sqrt2}{2}\), so \(2\sin45°\cos45°=2\left(\dfrac{\sqrt2}{2}\right)^2=1\) | 1 |
| \(\dfrac{1}{\tan60°}=\dfrac{1}{\sqrt3}=\boxed{\dfrac{\sqrt3}{3}}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
If \(\sec\theta=2\) and \(\theta\) is acute, determine \(\sin\theta\) without a calculator.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\cos\theta=\dfrac{1}{\sec\theta}=\dfrac{1}{2}\Rightarrow\theta=60°\) (special angle) | 1 |
| \(\sin60°=\boxed{\dfrac{\sqrt3}{2}}\) | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
A ladder leans against a wall, making an angle of \(65°\) with the ground. The foot of the ladder is \(2\)m from the wall. Determine the length of the ladder, correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\cos65^\circ=\dfrac{2}{L}\Rightarrow L=\dfrac{2}{\cos65^\circ}\). | 1 |
| \(L\approx\boxed{4{,}73\text{ m}}\). Round only this final value. | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
Solve for \(\theta\) (\(0°\leq\theta\leq90°\)): \(2\sin\theta-\sqrt2=0\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(2\sin\theta=\sqrt2\Rightarrow\sin\theta=\dfrac{\sqrt2}{2}\) | 1 |
| From the special-angle table, \(\sin45°=\dfrac{\sqrt2}{2}\) | 1 |
| \(\boxed{\theta=45°}\) | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
Determine, without a calculator: \(\dfrac{\sin^230°+\cos^260°}{\tan^245°}\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\sin30°=\cos60°=\dfrac12\), so \(\sin^230°=\cos^260°=\dfrac14\) | 1 |
| Numerator \(=\dfrac14+\dfrac14=\dfrac12\); \(\tan^245°=1^2=1\) | 1 |
| \(\boxed{\dfrac12}\) | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
A flagpole casts a shadow of length \(15\)m when the angle of elevation of the sun is \(38°\). Determine the height of the flagpole, correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(\tan38^\circ=\dfrac{h}{15}\Rightarrow h=15\tan38^\circ\). | 1 |
| \(h\approx\boxed{11{,}72\text{ m}}\). | 1 |
| Total | 2 |
Suggested practice allocation only. These marks are not copied from an official memo.
From the top of a vertical cliff \(45\)m high, the angle of depression to a boat out at sea is \(22°\). Determine the horizontal distance from the base of the cliff to the boat, correct to 2 decimal places.
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| The angle of elevation from the boat equals the 22-degree angle of depression: the horizontal lines are parallel. | 1 |
| \(\tan22^\circ=\frac{45}{d}\quad\Rightarrow\quad d=\frac{45}{\tan22^\circ}\) | 1 |
| \(d\approx111.38\text{ m}\) (keep the calculator value of tan 22 degrees until the last step). | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
Without using a calculator, determine the value of \(\cos150°\), and hence write down the value of \(\cos330°\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(150°\) is in Q2, reference angle \(=180°-150°=30°\); cos is negative in Q2: \(\cos150°=-\cos30°=\boxed{-\dfrac{\sqrt3}{2}}\) | 1 |
| \(330°\) is in Q4, reference angle \(=360°-330°=30°\); cos is positive in Q4: \(\cos330°=\cos30°=\boxed{\dfrac{\sqrt3}{2}}\) | 1 |
| Both angles share the same reference angle \(30°\) — only the sign, decided by CAST, differs. | 1 |
| Total | 3 |
Suggested practice allocation only. These marks are not copied from an official memo.
Without using a calculator, determine the value of \(\sin225°\) and the value of \(\cos225°\).
Equation Station practice, not an official exam question.
| Working and reasons | Practice marks |
|---|---|
| \(225°\) is in Q3, reference angle \(=225°-180°=45°\) | 1 |
| In Q3, only tan is positive (CAST) — so both sin and cos are negative | 1 |
| \(\sin225°=-\sin45°=\boxed{-\dfrac{\sqrt2}{2}}\) | 1 |
| \(\cos225°=-\cos45°=\boxed{-\dfrac{\sqrt2}{2}}\) | 1 |
| Total | 4 |
Suggested practice allocation only. These marks are not copied from an official memo.
2 questions
If \(4\tan\theta=-3\) and \(\cos\theta\geq0\), without using a calculator, determine: (a) \(\sin\theta\), (b) \(10\cos^2\theta\), (c) \(3\cot\theta+5\sin\theta\).
KZN November 2024, Q4. Checked against marking-guideline pages 4-5. The CAST explanation is support, not an extra mark.
| Working and reasons | Marks |
|---|---|
| CAST: tangent is negative and cosine is non-negative, so \(\theta\) is in Quadrant IV. Cosine cannot be zero because the given tangent is defined. | Support |
| \(4\tan\theta=-3\Rightarrow\tan\theta=-\dfrac34\). Choose \(x=4\), \(y=-3\). | 1 |
| \(r=\sqrt{4^2+(-3)^2}=5\). | 1 |
| 4.1 \(\sin\theta=\dfrac{y}{r}=\boxed{-\dfrac35}\). | 1 |
| 4.2 \(10\cos^2\theta=10\left(\dfrac45\right)^2\). | 1 |
| \(10\cos^2\theta=\boxed{\dfrac{32}{5}}\). | 1 |
| 4.3 \(3\cot\theta+5\sin\theta=3\left(-\dfrac43\right)+5\left(-\dfrac35\right)\). | 1 |
| \(=-4-3\). | 1 |
| \(=\boxed{-7}\). | 1 |
| Total | 8 |
A ladder \(AB\), \(10\)m long, leans against a vertical wall. The foot of the ladder \(B\) is \(4\)m from the base of the wall \(C\). (a) Calculate the height \(AC\) that the ladder reaches up the wall. (b) Calculate the angle of elevation of the ladder, \(\angle ABC\). (c) If the ladder is moved so that it now makes an angle of \(75°\) with the ground, calculate how high up the wall it will now reach.
WC Overberg November 2025, Q4. Question, answer and marking points checked against the supplied marking guideline.
| Working and reasons | Marks |
|---|---|
| (a) Pythagoras: \(AC^2=10^2-4^2=84\Rightarrow AC=\boxed{9{,}17\text{ m}}\) | 2 |
| (b) \(\cos(\angle ABC)=\dfrac{4}{10}=0{,}4\Rightarrow\angle ABC=\cos^{-1}(0{,}4)=\boxed{66{,}42°}\) | 3 |
| (c) Same ladder length, new angle: \(h=10\sin75^\circ\approx\boxed{9{,}66\text{ m}}\). | 3 |
| Total | 8 |