DP Mathematics HL Questionbank
Topic 3 - Core: Circular functions and trigonometry
Description
The aims of this topic are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry. On examination papers, radian measure should be assumed unless otherwise indicated, for example, by \(x \mapsto \sin {x^0}\) .
Directly related questions
- 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.TZ2.9: Show that \(\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \sec 2A + \tan 2A\) .
- 12N.2.hl.TZ0.12a: If \(\alpha \) is the angle between [CD] and the wall, show that...
- 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.TZ2.2: The depth, h(t) metres, of water at the entrance to a harbour at t hours after midnight on a...
- 13M.2.hl.TZ1.7a: Find, in terms of x, an expression for the cost of laying the cable.
- 11N.1.hl.TZ0.1: From a vertex of an equilateral triangle of side \(2x\), a circular arc is drawn to divide the...
- 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...
- 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].
- 11M.1.hl.TZ1.13e: By considering the solutions of the equation \(\cos 5\theta = 0\) , show that...
- 15M.1.hl.TZ2.6a: Show that length \({\text{AB}} = \frac{3}{{\sqrt 3 \cos \theta + \sin \theta }}\).
- 14N.1.hl.TZ0.13b: (i) Use the double angle identity...
- 15N.1.hl.TZ0.8a: Show that \(\sin \left( {\theta + \frac{\pi }{2}} \right) = \cos \theta \).
- 17M.2.hl.TZ1.8a: Find an expression for the volume of water \(V{\text{ }}({{\text{m}}^3})\) in the trough in terms...
- 17M.2.hl.TZ1.10a: Use the cosine rule to show that \({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\).
- 16N.2.hl.TZ0.9c: Hence find the value of \(r\) given that the shaded area is equal to 4.
- 18M.1.hl.TZ1.11b: Show that the area of the quadrilateral Q is \(\frac{{21\sqrt 3 }}{2}\).
- 18M.2.hl.TZ2.4a.ii: Find \({\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}\) in radians.
- 12M.1.hl.TZ1.10a: Show that cos \(\hat A - \sin \hat A = \frac{1}{{\sqrt 2 }}\).
- 12N.1.hl.TZ0.1: Given that \(\frac{\pi }{2} < \alpha < \pi \) and \(\cos \alpha = - \frac{3}{4}\), find...
- 08M.2.hl.TZ1.7: A system of equations is given by \[\cos x + \cos y =...
- 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.2.hl.TZ2.8: The vertices of an equilateral triangle, with perimeter P and area A , lie on a circle with...
- SPNone.2.hl.TZ0.7b: the time at which the interception occurs, correct to the nearest minute.
- 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...
- 13M.1.hl.TZ1.12f: Prove that \(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}} \).
- 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.
- 11M.1.hl.TZ1.13d: Hence solve the equation \(\cos 5\theta + \cos 3\theta + \cos \theta = 0\) , where...
- 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...
- 09M.2.hl.TZ2.12: (a) Explain why \(x < 40\) . (b) Show that cosθ = x −10 50. (c) (i) Find an...
- 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.TZ1.5a: Use the identity \(\cos 2\theta = 2{\cos ^2}\theta - 1\) to prove that...
- 15M.1.hl.TZ2.3: Find all solutions to the equation \(\tan x + \tan 2x = 0\) where \(0^\circ \le x < 360^\circ\).
- 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...
- 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.TZ0.14a: Show that \(\sin (B + C) = \frac{1}{2}\).
- 15N.2.hl.TZ0.4a: Find the value of (i) \(A\); (ii) \(B\); (iii) \(C\).
- 17N.2.hl.TZ0.11b.ii: Express \(\sin 2x\) in terms of \(u\).
- 16N.1.hl.TZ0.13a: Find the value of...
- 16M.1.hl.TZ1.5c: Find BC in the form \(a + \sqrt b \) where \(a,{\text{ }}b \in \mathbb{Z}\).
- 16M.2.hl.TZ2.11b: Show that \(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\).
- 08M.1.hl.TZ2.3: In the diagram below, AD is perpendicular to BC. CD = 4, BD = 2 and AD = 3....
- 08N.1.hl.TZ0.11: (a) Sketch the curve \(f(x) = \sin 2x\) , \(0 \leqslant x \leqslant \pi \) . (b) Hence...
- 11M.1.hl.TZ2.7b: Find the area of triangle POT in terms of r and \(\theta \) .
- 09M.1.hl.TZ1.6: The diagram below shows two straight lines intersecting at O and two circles, each with centre O....
- 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.TZ0.1c: Find the value of \(\cos \left( {\frac{\theta }{2}} \right)\) , giving your answer in the form...
- 10M.2.hl.TZ1.1: The graph below shows \(y = a\cos (bx) + c\). Find the value of a, the value of b and...
- 10N.2.hl.TZ0.1: Triangle ABC has AB = 5 cm, BC = 6 cm and area 10 \({\text{c}}{{\text{m}}^2}\). (a) Find...
- 13M.1.hl.TZ2.10b: Hence find the value of...
- 11M.1.hl.TZ1.5b: Hence find the value of \(\cot \frac{\pi }{8}\) in the form \(a + b\sqrt 2 \) , where...
- 15M.1.hl.TZ1.1b: Find the total length of the perimeter of the logo.
- 15M.2.hl.TZ2.1a: Find the area of the triangle.
- 14N.2.hl.TZ0.10a: Show that the area, \(A\;{\text{c}}{{\text{m}}^2}\), of the triangle is given by...
- 17M.1.hl.TZ1.3: Solve the equation \({\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi \).
- 17M.1.hl.TZ2.11a: Solve...
- 16N.1.hl.TZ0.13b: Show that \(\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi \) where...
- 16N.2.hl.TZ0.5b: State the range of \(f\).
- 16M.1.hl.TZ1.3a: Write down the value of \(a\) and the value of \(d\).
- 16M.1.hl.TZ2.3a: Show that \(\cot \alpha = \tan \left( {\frac{\pi }{2} - \alpha } \right)\) for...
- 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
- 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.2.hl.TZ2.5: Consider triangle ABC with \({\rm{B}}\hat {\rm{A}}{\rm{C}} = 37.8^\circ \) , AB = 8.75 and BC = 6...
- 11M.2.hl.TZ2.1: The points P and Q lie on a circle, with centre O and radius 8 cm, such that...
- 10M.1.hl.TZ1.13: (a) Show that...
- 10M.2.hl.TZ2.5: Consider the triangle ABC where \({\rm{B\hat AC}} = 70^\circ \), AB = 8 cm and AC = 7 cm. The...
- 11M.2.hl.TZ1.3: Given \(\Delta \)ABC, with lengths shown in the diagram below, find the length of the line...
- 14M.1.hl.TZ2.5b: Solve \(\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}\) for...
- 13N.1.hl.TZ0.8: (a) Prove the trigonometric identity \(\sin (x + y)\sin (x - y) = {\sin ^2}x - {\sin...
- 15N.1.hl.TZ0.1: The following diagram shows a sector of a circle where \({\rm{A\hat OB}} = x\) radians and the...
- 17N.2.hl.TZ0.11b.iii: Hence show that \(f(x) = 0\) can be expressed as \({u^3} - 7{u^2} + 15u - 9 = 0\).
- 16N.2.hl.TZ0.5a: Sketch the graph of \(f\) indicating clearly any intercepts with the axes and the coordinates of...
- 18M.1.hl.TZ1.8: Let \(a = {\text{sin}}\,b,\,\,0 < b < \frac{\pi }{2}\). Find, in terms of b, the solutions...
- 18M.2.hl.TZ2.4b: Find the area of the shaded region.
- 12N.1.hl.TZ0.8b: Hence, show that \(\tan \theta = \frac{1}{{1 + 2\pi }}\), where \(\theta \) is the acute angle...
- 12N.1.hl.TZ0.7b: Determine the values of k for which the conditions above define a unique triangle.
- 09M.1.hl.TZ1.2: The diagram below shows a curve with equation \(y = 1 + k\sin x\) , defined for...
- SPNone.1.hl.TZ0.6a: Determine BC, giving your answer in the form \(k\sqrt 3 \), \(k \in {\mathbb{Z}^ + }\) .
- 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...
- 11M.1.hl.TZ1.5a: Show that \(\frac{{\sin 2\theta }}{{1 + \cos 2\theta }} = \tan \theta \) .
- 14M.2.hl.TZ1.5a: Find the coordinates of the points where the line meets the curve.
- 14M.2.hl.TZ2.4: The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the...
- 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...
- 16N.2.hl.TZ0.5c: Solve the inequality \(\left| {3x\arccos (x)} \right| > 1\).
- 16N.2.hl.TZ0.7b: Find the difference between the areas of the two possible triangles ABC.
- 16M.1.hl.TZ1.3c: Find the smallest possible value of \(c\), given \(c > 0\).
- 16M.1.hl.TZ1.5a: Expand and simplify \({\left( {1 - \sqrt 3 } \right)^2}\).
- 16M.2.hl.TZ2.1: ABCD is a quadrilateral where...
- 18M.2.hl.TZ2.4a.i: Find AM.
- 12M.1.hl.TZ1.5a: For what values of x does \(f(x)\) not exist?
- 12M.2.hl.TZ2.3b: Find the possible values of AC .
- 12N.1.hl.TZ0.7a: For the case k = 4 , find the two possible values of QR.
- SPNone.1.hl.TZ0.7a: Show that this system does not have a unique solution for any value of \(\lambda \) .
- SPNone.1.hl.TZ0.1a: Write down the value of \(\sin \theta \) .
- SPNone.1.hl.TZ0.6b: The point D lies on (BC), and (AD) is perpendicular to (BC). Determine AD.
- 11N.2.hl.TZ0.4b: Hence, or otherwise, solve the equation \(\arcsin x = \arctan a\).
- 09N.2.hl.TZ0.3: (a) Show that the area of the shaded region is \(8\sin x - 2x\) . (b) Find the maximum...
- 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.TZ1.12c: Using (a) (ii) and your answer from (b) show that...
- 15M.2.hl.TZ2.1b: Find \(AC\).
- 14N.2.hl.TZ0.14b: Robert conjectures that \({\rm{C\hat AB}}\) can have two possible values. Show that Robert’s...
- 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.
- 16N.2.hl.TZ0.9a: Find an expression for the shaded area in terms of \(\alpha \), \(\theta \) and \(r\).
- 16M.1.hl.TZ1.5b: By writing \(15^\circ \) as \(60^\circ - 45^\circ \) find the value of \(\cos (15^\circ )\).
- 12M.1.hl.TZ1.5b: Simplify the expression \(\frac{{\sin 3x}}{{\sin x}} - \frac{{\cos 3x}}{{\cos x}}\).
- 08M.1.hl.TZ1.4: In triangle ABC, AB = 9 cm , AC = 12 cm , and \(\hat B\) is twice the size of \({\hat C}\)...
- 11M.1.hl.TZ2.7a: Find the area of triangle AOP in terms of r and \(\theta \) .
- 11M.1.hl.TZ2.13a: (i) Sketch the graphs of \(y = \sin x\) and \(y = \sin 2x\) , on the same set of axes, for...
- 09M.1.hl.TZ1.5: (a) Show that...
- SPNone.1.hl.TZ0.1b: Find the value of \(\tan 2\theta \) .
- SPNone.2.hl.TZ0.7a: the value of \(\theta \).
- 13M.1.hl.TZ1.10a: Find all values of x for \(0.1 \leqslant x \leqslant 1\) such that \(\sin (\pi {x^{ - 1}}) = 0\).
- 10M.1.hl.TZ2.6: If x satisfies the equation...
- 11N.1.hl.TZ0.4b: show that \({\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x\);
- 11N.1.hl.TZ0.8a: Show that \(t = \frac{2}{3}(2\cos \theta + \theta )\).
- 11N.2.hl.TZ0.4a: Given that...
- 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...
- 15M.1.hl.TZ1.1a: Find, in radians, the value of the angle \(\theta \), as indicated on the diagram.
- 15M.2.hl.TZ1.4: A triangle \(ABC\) has \(\hat A = 50^\circ \), \({\text{AB}} = 7{\text{ cm}}\) and...
- 15N.2.hl.TZ0.7: Triangle \(ABC\) has area \({\text{21 c}}{{\text{m}}^{\text{2}}}\). The sides \(AB\) and \(AC\)...
- 17M.1.hl.TZ2.11b: Show that \(\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}\).
- 17N.2.hl.TZ0.11c: Solve the equation \(f(x) = 0\), giving your answers in the form \(\arctan k\) where...
- 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.TZ2.9: 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...
- 08M.1.hl.TZ2.9: The diagram below shows the boundary of the cross-section of a water channel. The...
- 11M.1.hl.TZ2.7c: Using your results from part (a) and part (b), show that...
- SPNone.1.hl.TZ0.8: The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a \( \times \) b = 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.TZ2.13: The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas...
- 13M.1.hl.TZ2.10a: Given that...
- 13M.2.hl.TZ2.6a: Solve the equation \(3{\cos ^2}x - 8\cos x + 4 = 0\), where...
- 13M.2.hl.TZ2.13b: (i) Calculate the value of \(\theta \) when x = 0. (ii) Calculate the value of...
- 13M.2.hl.TZ2.6b: Find the exact values of \(\sec x\) satisfying the equation \(3{\sec ^4}x - 8{\sec ^2}x + 4 = 0\).
- 13N.2.hl.TZ0.8a: Show that the shaded area can be expressed as \(50\theta - 50\sin \theta \).
- 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.TZ0.9: Solve the equation \(\sin 2x - \cos 2x = 1 + \sin x - \cos x\) for...
- 15N.2.hl.TZ0.4b: Solve \(f(x) = 3\) for \(0 \le x \le \pi \).
- 17M.2.hl.TZ1.10b: Calculate the two corresponding values of PQ.
- 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...
- 17N.2.hl.TZ0.3: This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius...
- 17N.2.hl.TZ0.11b.i: Express \(\sin x\) in terms of \(\mu \).
- 16N.1.hl.TZ0.13d: Hence or otherwise solve the equation \(\sin x + \sin 3x = \cos x\) in the interval...
- 16N.2.hl.TZ0.9b: Show that \(\alpha = 4\arcsin \frac{1}{4}\).
- 16N.2.hl.TZ0.7a: Use the cosine rule to find the two possible values for AC.
- 16M.1.hl.TZ1.3b: Find the value of \(b\).
- 16M.2.hl.TZ1.4: The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The...
- 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 \(0 < x < \frac{\pi }{2}\).
- 16M.2.hl.TZ2.11a: Find the value of \(\alpha \) when \(x = 10\).
- 18M.2.hl.TZ1.3: Let ...
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 \(\frac{{21\sqrt 3 }}{2}\).
