DP Mathematics HL Questionbank

• Syllabus

Topic 3 - Core: Circular functions and trigonometry

Sub sections and their related questions

3.1

• 12M.2.hl.TZ2.9: Two discs, one of radius 8 cm and one of radius 5 cm, are placed such that they touch each other....
• 08M.1.hl.TZ1.3: A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence. The angle...
• 11M.1.hl.TZ2.7c: Using your results from part (a) and part (b), show that...
• 11M.2.hl.TZ2.1: The points P and Q lie on a circle, with centre O and radius 8 cm, such that...
• 09M.1.hl.TZ1.6: The diagram below shows two straight lines intersecting at O and two circles, each with centre O....
• 13M.2.hl.TZ1.5: A rectangle is drawn around a sector of a circle as shown. If the angle of the sector is 1 radian...
• 10M.2.hl.TZ2.13: The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas...
• 13M.2.hl.TZ2.1a: Find ${\rm{A\hat OB}}$, expressing your answer in radians correct to four significant figures.
• 13M.2.hl.TZ2.1b: Determine the area of the shaded region.
• 11N.1.hl.TZ0.1: From a vertex of an equilateral triangle of side $2x$, a circular arc is drawn to divide the...
• 11N.1.hl.TZ0.8a: Show that $t = \frac{2}{3}(2\cos \theta + \theta )$.
• 11M.2.hl.TZ1.6: The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5...
• 09N.2.hl.TZ0.3: (a)     Show that the area of the shaded region is $8\sin x - 2x$ . (b)     Find the maximum...
• 09M.2.hl.TZ2.12: (a)     Explain why $x < 40$ . (b)     Show that cosθ = x −10 50. (c)     (i)     Find an...
• 14M.2.hl.TZ2.4: The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the...
• 13N.2.hl.TZ0.8a: Show that the shaded area can be expressed as $50\theta  - 50\sin \theta$.
• 15M.1.hl.TZ1.1a: Find, in radians, the value of the angle $\theta$, as indicated on the diagram.
• 15M.1.hl.TZ1.1b: Find the total length of the perimeter of the logo.
• 15M.2.hl.TZ2.8a: Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to...
• 15M.2.hl.TZ2.8b: Bill replaces Gruff’s rope with another, this time of length $a,{\text{ }}4 < a < 10$, so...
• 15N.1.hl.TZ0.1: The following diagram shows a sector of a circle where ${\rm{A\hat OB}} = x$ radians and the...
• 16M.2.hl.TZ1.4: The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The...
• 16N.2.hl.TZ0.9a: Find an expression for the shaded area in terms of $\alpha$, $\theta$ and $r$.
• 16N.2.hl.TZ0.9b: Show that $\alpha  = 4\arcsin \frac{1}{4}$.
• 16N.2.hl.TZ0.9c: Hence find the value of $r$ given that the shaded area is equal to 4.
• 17M.2.hl.TZ1.8a: Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms...
• 17N.2.hl.TZ0.3: This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius...
• 18M.2.hl.TZ2.4a.i: Find AM.
• 18M.2.hl.TZ2.4a.ii: Find ${\text{A}}\mathop {\text{M}}\limits^ \wedge  {\text{P}}$ in radians.
• 18M.2.hl.TZ2.4b: Find the area of the shaded region.

3.2

• 12M.1.hl.TZ1.5a: For what values of x does $f(x)$ not exist?
• 12M.1.hl.TZ1.10a: Show that cos $\hat A - \sin \hat A = \frac{1}{{\sqrt 2 }}$.
• 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
• 12M.1.hl.TZ2.9: Show that $\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \sec 2A + \tan 2A$ .
• 12M.1.hl.TZ2.10a: State the two zeros of f .
• 12M.2.hl.TZ2.3a: By drawing a diagram, show why there are two triangles consistent with this information.
• 12M.2.hl.TZ2.3b: Find the possible values of AC .
• 12N.1.hl.TZ0.1: Given that $\frac{\pi }{2} < \alpha  < \pi$ and $\cos \alpha  = - \frac{3}{4}$, find...
• 12N.2.hl.TZ0.12a: If $\alpha$ is the angle between [CD] and the wall, show that...
• 11M.1.hl.TZ2.13a: (i)     Sketch the graphs of $y = \sin x$ and $y = \sin 2x$ , on the same set of axes, for...
• SPNone.1.hl.TZ0.1a: Write down the value of $\sin \theta$ .
• 13M.1.hl.TZ1.10a: Find all values of x for $0.1 \leqslant x \leqslant 1$ such that $\sin (\pi {x^{ - 1}}) = 0$.
• 13M.2.hl.TZ2.13a: Find an expression for $\theta$ in terms of x, where x is the distance of P from the base of...
• 13M.2.hl.TZ2.13b: (i)     Calculate the value of $\theta$ when x = 0. (ii)     Calculate the value of...
• 11N.1.hl.TZ0.8a: Show that $t = \frac{2}{3}(2\cos \theta + \theta )$.
• 12M.1.hl.TZ1.10b: By squaring both sides of the equation in part (a), solve the equation to find the angles in the...
• 12M.1.hl.TZ1.10c: Apply Pythagoras’ theorem in the triangle ABC to find BC, and hence show that...
• 12M.1.hl.TZ1.10d: Hence, or otherwise, calculate the length of the perpendicular from B to [AC].
• 13N.1.hl.TZ0.8: (a)     Prove the trigonometric identity \(\sin (x + y)\sin (x - y) = {\sin ^2}x - {\sin...
• 14N.2.hl.TZ0.14a: Show that $\sin (B + C) = \frac{1}{2}$.
• 15M.1.hl.TZ2.11c: Let $y = g \circ f(x)$, find an exact value for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ at the...
• 15M.2.hl.TZ1.12c: Using (a) (ii) and your answer from (b) show that...
• 15N.1.hl.TZ0.8a: Show that $\sin \left( {\theta  + \frac{\pi }{2}} \right) = \cos \theta$.
• 16M.1.hl.TZ2.3a: Show that $\cot \alpha  = \tan \left( {\frac{\pi }{2} - \alpha } \right)$ for...
• 16M.2.hl.TZ2.11a: Find the value of $\alpha$ when $x = 10$.
• 16M.2.hl.TZ2.11b: Show that $\tan \alpha  = \frac{{2x}}{{{x^2} + 35}}$.
• 16N.1.hl.TZ0.13a: Find the value of...
• 17M.1.hl.TZ1.3: Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$.

3.3

• 12M.1.hl.TZ1.5b: Simplify the expression $\frac{{\sin 3x}}{{\sin x}} - \frac{{\cos 3x}}{{\cos x}}$.
• 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
• 12M.1.hl.TZ2.9: Show that $\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \sec 2A + \tan 2A$ .
• 12N.1.hl.TZ0.1: Given that $\frac{\pi }{2} < \alpha  < \pi$ and $\cos \alpha  = - \frac{3}{4}$, find...
• 12N.1.hl.TZ0.8b: Hence, show that $\tan \theta  = \frac{1}{{1 + 2\pi }}$, where $\theta$ is the acute angle...
• 08M.1.hl.TZ2.3: In the diagram below, AD is perpendicular to BC. CD  =  4, BD  =  2 and AD  =  3....
• SPNone.1.hl.TZ0.1b: Find the value of $\tan 2\theta$ .
• SPNone.1.hl.TZ0.1c: Find the value of $\cos \left( {\frac{\theta }{2}} \right)$ , giving your answer in the form...
• 13M.1.hl.TZ1.12f: Prove that $\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}}$.
• 10M.1.hl.TZ1.13: (a)     Show that...
• 10M.1.hl.TZ2.6: If x satisfies the equation...
• 13M.1.hl.TZ2.10a: Given that...
• 13M.1.hl.TZ2.10b: Hence find the value of...
• 11N.1.hl.TZ0.4b: show that ${\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x$;
• 11N.2.hl.TZ0.4a: Given that...
• 12M.1.hl.TZ1.10b: By squaring both sides of the equation in part (a), solve the equation to find the angles in the...
• 11M.1.hl.TZ1.5a: Show that $\frac{{\sin 2\theta }}{{1 + \cos 2\theta }} = \tan \theta$ .
• 11M.1.hl.TZ1.5b: Hence find the value of $\cot \frac{\pi }{8}$ in the form $a + b\sqrt 2$ , where...
• 14M.1.hl.TZ1.5b: Find a similar expression for $\sin \frac{1}{2}x,{\text{ }}0 \leqslant x \leqslant \pi$.
• 14M.1.hl.TZ1.10: Given that $\sin x + \cos x = \frac{2}{3}$, find $\cos 4x$.
• 14M.1.hl.TZ2.9: The first three terms of a geometric sequence are $\sin x,{\text{ }}\sin 2x$ and...
• 13N.1.hl.TZ0.8: (a)     Prove the trigonometric identity \(\sin (x + y)\sin (x - y) = {\sin ^2}x - {\sin...
• 14M.1.hl.TZ1.5a: Use the identity $\cos 2\theta  = 2{\cos ^2}\theta  - 1$ to prove that...
• 14N.1.hl.TZ0.13b: (i)     Use the double angle identity...
• 14N.2.hl.TZ0.14a: Show that $\sin (B + C) = \frac{1}{2}$.
• 15N.1.hl.TZ0.8a: Show that $\sin \left( {\theta  + \frac{\pi }{2}} \right) = \cos \theta$.
• 15N.1.hl.TZ0.9: Solve the equation $\sin 2x - \cos 2x = 1 + \sin x - \cos x$ for...
• 16M.1.hl.TZ1.5b: By writing $15^\circ$ as $60^\circ  - 45^\circ$ find the value of $\cos (15^\circ )$.
• 16N.1.hl.TZ0.13b: Show that $\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi$ where...
• 17M.1.hl.TZ2.11a: Solve...
• 17M.1.hl.TZ2.11b: Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}$.
• 18M.2.hl.TZ1.3: Let ...

3.4

• 08M.2.hl.TZ2.2: The depth, h(t) metres, of water at the entrance to a harbour at t hours after midnight on a...
• 09M.1.hl.TZ1.2: The diagram below shows a curve with equation $y = 1 + k\sin x$ , defined for...
• 10M.2.hl.TZ1.1: The graph below shows $y = a\cos (bx) + c$.     Find the value of a, the value of b and...
• 15N.2.hl.TZ0.4a: Find the value of (i)     $A$; (ii)     $B$; (iii)     $C$.
• 16M.1.hl.TZ1.3a: Write down the value of $a$ and the value of $d$.
• 16M.1.hl.TZ1.3b: Find the value of $b$.
• 16M.1.hl.TZ1.3c: Find the smallest possible value of $c$, given $c > 0$.

3.5

• 08M.1.hl.TZ2.9: The diagram below shows the boundary of the cross-section of a water channel.     The...
• 09M.1.hl.TZ1.5: (a)     Show that...
• 14M.1.hl.TZ2.9: The first three terms of a geometric sequence are $\sin x,{\text{ }}\sin 2x$ and...
• 14M.1.hl.TZ2.14a: Sketch the graph of $y = h(x)$.
• 15M.2.hl.TZ2.8b: Bill replaces Gruff’s rope with another, this time of length $a,{\text{ }}4 < a < 10$, so...
• 16N.2.hl.TZ0.5a: Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of...
• 16N.2.hl.TZ0.5b: State the range of $f$.
• 16N.2.hl.TZ0.5c: Solve the inequality $\left| {3x\arccos (x)} \right| > 1$.

3.6

• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.2.hl.TZ1.11b: The points ${\text{P}}(p,{\text{ }}3)$ and ${\text{Q}}(q,{\text{ }}3){\text{, }}q > p$,...
• 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
• 08M.2.hl.TZ1.7: A system of equations is given by \[\cos x + \cos y =...
• 08N.1.hl.TZ0.11: (a)     Sketch the curve $f(x) = \sin 2x$ , $0 \leqslant x \leqslant \pi$ . (b)     Hence...
• 11M.1.hl.TZ2.13a: (i)     Sketch the graphs of $y = \sin x$ and $y = \sin 2x$ , on the same set of axes, for...
• 10M.1.hl.TZ1.13: (a)     Show that...
• 13M.2.hl.TZ2.6a: Solve the equation $3{\cos ^2}x - 8\cos x + 4 = 0$, where...
• 13M.2.hl.TZ2.6b: Find the exact values of $\sec x$ satisfying the equation $3{\sec ^4}x - 8{\sec ^2}x + 4 = 0$.
• 11N.2.hl.TZ0.4b: Hence, or otherwise, solve the equation $\arcsin x = \arctan a$.
• 12M.1.hl.TZ1.10b: By squaring both sides of the equation in part (a), solve the equation to find the angles in the...
• 11M.1.hl.TZ1.13d: Hence solve the equation $\cos 5\theta + \cos 3\theta + \cos \theta = 0$ , where...
• 11M.1.hl.TZ1.13e: By considering the solutions of the equation $\cos 5\theta = 0$ , show that...
• 14M.2.hl.TZ1.5a: Find the coordinates of the points where the line meets the curve.
• 14M.1.hl.TZ2.5b: Solve $\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}$ for...
• 14N.2.hl.TZ0.14b: Robert conjectures that ${\rm{C\hat AB}}$ can have two possible values. Show that Robert’s...
• 15M.1.hl.TZ2.3: Find all solutions to the equation $\tan x + \tan 2x = 0$ where $0^\circ  \le x < 360^\circ$.
• 15N.1.hl.TZ0.9: Solve the equation $\sin 2x - \cos 2x = 1 + \sin x - \cos x$ for...
• 15N.2.hl.TZ0.4b: Solve $f(x) = 3$ for $0 \le x \le \pi$.
• 16M.1.hl.TZ2.9a: verify that $x = \frac{\pi }{{12}}$ is a solution to the equation;
• 16M.1.hl.TZ2.9b: hence find the other solution to the equation for $0 < x < \frac{\pi }{2}$.
• 16N.1.hl.TZ0.13d: Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval...
• 17M.1.hl.TZ1.3: Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$.
• 17N.2.hl.TZ0.11b.i: Express $\sin x$ in terms of $\mu$.
• 17N.2.hl.TZ0.11b.ii: Express $\sin 2x$ in terms of $u$.
• 17N.2.hl.TZ0.11b.iii: Hence show that $f(x) = 0$ can be expressed as ${u^3} - 7{u^2} + 15u - 9 = 0$.
• 17N.2.hl.TZ0.11c: Solve the equation $f(x) = 0$, giving your answers in the form $\arctan k$ where...
• 18M.1.hl.TZ1.8: Let $a = {\text{sin}}\,b,\,\,0 < b < \frac{\pi }{2}$. Find, in terms of b, the solutions...

3.7

• 12N.1.hl.TZ0.7a: For the case k = 4 , find the two possible values of QR.
• 12N.1.hl.TZ0.7b: Determine the values of k for which the conditions above define a unique triangle.
• 12N.2.hl.TZ0.13e: Let Y be a point on ${L_1}$ and Z be a point on ${L_2}$ such that XY is perpendicular to YZ...
• 08M.1.hl.TZ1.4: In triangle ABC, AB = 9 cm , AC = 12 cm , and $\hat B$ is twice the size of ${\hat C}$...
• 08M.2.hl.TZ1.12Part A: In triangle ABC, BC = a , AC = b , AB = c and [BD] is perpendicular to [AC].   (a)     Show...
• 08M.2.hl.TZ2.5: Consider triangle ABC with ${\rm{B}}\hat {\rm{A}}{\rm{C}} = 37.8^\circ$ , AB = 8.75 and BC = 6...
• 08N.2.hl.TZ0.1: In a triangle ABC, $\hat A = 35^\circ$, BC = 4 cm and AC = 6.5 cm. Find the possible values of...
• 11M.1.hl.TZ2.7a: Find the area of triangle AOP in terms of r and $\theta$ .
• 11M.1.hl.TZ2.7b: Find the area of triangle POT in terms of r and $\theta$ .
• 11M.2.hl.TZ2.8: The vertices of an equilateral triangle, with perimeter P and area A , lie on a circle with...
• 09M.1.hl.TZ2.9: A triangle has sides of length $({n^2} + n + 1)$, $(2n + 1)$ and $({n^2} - 1)$ where...
• SPNone.1.hl.TZ0.6a: Determine BC, giving your answer in the form $k\sqrt 3$, $k \in {\mathbb{Z}^ + }$ .
• SPNone.1.hl.TZ0.6b: The point D lies on (BC), and (AD) is perpendicular to (BC). Determine AD.
• SPNone.1.hl.TZ0.7a: Show that this system does not have a unique solution for any value of $\lambda$ .
• SPNone.1.hl.TZ0.8: The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a $\times$ b = b...
• SPNone.2.hl.TZ0.7a: the value of $\theta$.
• SPNone.2.hl.TZ0.7b: the time at which the interception occurs, correct to the nearest minute.
• 13M.2.hl.TZ1.7a: Find, in terms of x, an expression for the cost of laying the cable.
• 10M.2.hl.TZ1.3: In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B...
• 10M.2.hl.TZ1.13: Points A, B and C are on the circumference of a circle, centre O and radius $r$ . A trapezium...
• 10M.2.hl.TZ2.5: Consider the triangle ABC where ${\rm{B\hat AC}} = 70^\circ$, AB = 8 cm and AC = 7 cm. The...
• 10N.2.hl.TZ0.1: Triangle ABC has AB = 5 cm, BC = 6 cm and area 10 ${\text{c}}{{\text{m}}^2}$. (a)     Find...
• 11M.2.hl.TZ1.3: Given $\Delta$ABC, with lengths shown in the diagram below, find the length of the line...
• 14M.1.hl.TZ1.7: The triangle ABC is equilateral of side 3 cm. The point D lies on [BC] such that BD = 1 cm. Find...
• 14N.2.hl.TZ0.9a: If $n > 2$ and even, show that $C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}$.
• 14N.2.hl.TZ0.10a: Show that the area, $A\;{\text{c}}{{\text{m}}^2}$, of the triangle is given by...
• 15M.1.hl.TZ2.6a: Show that length ${\text{AB}} = \frac{3}{{\sqrt 3 \cos \theta  + \sin \theta }}$.
• 15M.2.hl.TZ1.4: A triangle $ABC$ has $\hat A = 50^\circ$, ${\text{AB}} = 7{\text{ cm}}$ and...
• 15M.2.hl.TZ2.1a: Find the area of the triangle.
• 15M.2.hl.TZ2.1b: Find $AC$.
• 15N.2.hl.TZ0.7: Triangle $ABC$ has area ${\text{21 c}}{{\text{m}}^{\text{2}}}$. The sides $AB$ and $AC$...
• 16M.1.hl.TZ1.5a: Expand and simplify ${\left( {1 - \sqrt 3 } \right)^2}$.
• 16M.1.hl.TZ1.5c: Find BC in the form $a + \sqrt b$ where $a,{\text{ }}b \in \mathbb{Z}$.
• 16M.2.hl.TZ2.1: ABCD is a quadrilateral where...
• 16N.2.hl.TZ0.7a: Use the cosine rule to find the two possible values for AC.
• 16N.2.hl.TZ0.7b: Find the difference between the areas of the two possible triangles ABC.
• 17M.2.hl.TZ1.10a: Use the cosine rule to show that ${r^2} - 12\sqrt 3 r + 144 - {p^2} = 0$.
• 17M.2.hl.TZ1.10b: Calculate the two corresponding values of PQ.
• 17M.2.hl.TZ1.10c: Hence, find the area of the smaller triangle.
• 17M.2.hl.TZ1.10d: Determine the range of values of $p$ for which it is possible to form two triangles.
• 17M.2.hl.TZ2.4b: The triangle ABC is shown in the following diagram. Given that $\cos B < \frac{1}{4}$, find...
• 17N.2.hl.TZ0.5: Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the...
• 18M.1.hl.TZ1.11b: Show that the area of the quadrilateral Q is $\frac{{21\sqrt 3 }}{2}$.