IB Questionbank

User interface language: English | Español

Date	May 2018	Marks available	5	Reference code	18M.2.AHL.TZ1.H_3
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Express	Question number	H_3	Adapted from	N/A

Question

Let $f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)$ where $0 < x < \frac{\pi }{2}$ .

Express $f\left( x \right)$ in terms of sin $x$ and cos $x$ .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

${\text{tan}}\left( {x + \pi } \right) = \tan x\left( { = \frac{{{\text{sin}}\,x}}{{{\text{cos}}\,x}}} \right)$ (M1)A1

${\text{cos}}\left( {x - \frac{\pi }{2}} \right) = {\text{sin}}\,x$ (M1)A1

Note: The two M1s can be awarded for observation or for expanding.

${\text{tan}}\left( {x + \pi } \right) = {\text{cos}}\left( {x - \frac{\pi }{2}} \right) = \frac{{{\text{si}}{{\text{n}}^2}\,x}}{{{\text{cos}}\,x}}$ A1

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3— Geometry and trigonometry » AHL 3.10—Compound angle identities

Show 39 related questions

EXM.3.AHL.TZ0.5a.iii:
Write down the period of this graph
EXM.3.AHL.TZ0.5f.ii:
Show that $\frac{a}{b} = {\text{tan}}\,\theta$ .
18N.2.AHL.TZ0.H_11a:
Find an expression for ${\theta _p}$ in terms of $p$ .
18N.2.AHL.TZ0.H_11b:
Show that $p = \frac{{{q^2} + q - 1}}{{2q + 1}}$ .
EXM.3.AHL.TZ0.5c.ii:
Comment on your answer to part (c)(i).
EXM.3.AHL.TZ0.5e.i:
$A$ .
EXM.3.AHL.TZ0.5b.ii:
Find the value of $C$ .
EXM.3.AHL.TZ0.5e.iii:
$C$ .
SPM.2.AHL.TZ0.12b:
Verify that $x = {\text{tan}}\,\theta$ and $x = - \,{\text{cot}}\,\theta$ satisfy the equation ${x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0$ .
18N.2.AHL.TZ0.H_11c:
By sketching the graph of $p$ as a function of $q$ , determine the range of values of $p$ for which there are possible values of $q$ .
19M.1.AHL.TZ1.H_9c:
Hence or otherwise find $\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x$ in the form ${\text{ln}}\left( {a + \sqrt b } \right)$ where $a$ , $b \in \mathbb{Z}$ .
EXM.3.AHL.TZ0.5a.i:
Sketch the graph $y = 3\,{\text{sin}}\,x + 4\,{\text{cos}}\,x$ , for $- 2\pi \leqslant x \leqslant 2\pi$
19M.1.AHL.TZ1.H_9a:
Show that ${\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x$ .
EXM.3.AHL.TZ0.5c.i:
Find ${\text{arctan}}\frac{3}{4}$ , giving the answer to 3 significant figures.
EXM.3.AHL.TZ0.5g:
Hence prove your conjectures in part (e).
EXM.3.AHL.TZ0.5b.i:
Use your answers from part (a) to write down the value of $A$ , $B$ and $D$ .
SPM.2.AHL.TZ0.12d:
Using the results from parts (b) and (c) find the exact value of ${\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}$ .

Give your answer in the form $a + b\sqrt 3$ where $a$ , $b \in \mathbb{Z}$ .
EXM.3.AHL.TZ0.5f.i:
Show that $\frac{b}{{\sqrt {{a^2} + {b^2}} }} = {\text{cos}}\,\theta$ .
18N.1.AHL.TZ0.H_11a:
Find the roots of ${z^{24}} = 1$ which satisfy the condition $0 < {\text{arg}}\left( z \right) < \frac{\pi }{2}$ , expressing your answers in the form $r{e^{{\text{i}}\theta }}$ , where $r$ , $\theta \in {\mathbb{R}^ + }$ .
16N.1.AHL.TZ0.H_13d:
Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .
19M.1.AHL.TZ1.H_9b:
Show that ${\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}$ .
21M.1.AHL.TZ2.12b:
Show that $arctan p + arctan q \equiv arctan (\frac{p + q}{1 - p q})$ where $p, q > 0$ and $p q < 1$ .
EXN.1.AHL.TZ0.7:
Consider quadrilateral $PQRS$ where $[PQ]$ is parallel to $[SR]$ .

In $PQRS$ , $PQ = x$ , $SR = y$ , $R \hat{S} P = α$ and $Q \hat{R} S = β$ .

Find an expression for $PS$ in terms of $x, y, \sin β$ and $\sin (α + β)$ .
21M.1.AHL.TZ2.12c:
Verify that $arctan (2 x + 1) = arctan (\frac{x}{x + 1}) + \frac{π}{4}$ for $x \in ℝ, x > 0$ .
EXM.3.AHL.TZ0.5a.ii:
Write down the amplitude of this graph
17M.1.AHL.TZ2.H_11c.i:
Find the modulus and argument of $z$ in terms of $\theta$ . Express each answer in its simplest form.
EXM.3.AHL.TZ0.5d:
By considering the graph of $y = 5\,{\text{sin}}\,x + 12\,{\text{cos}}\,x$ , find the value of $A$ , $B$ , $C$ and $D$ .
18N.1.AHL.TZ0.H_11b.i:
Show that Re S = Im S.
SPM.2.AHL.TZ0.12c:
Hence, or otherwise, show that the exact value of ${\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3$ .
EXM.3.AHL.TZ0.5e.ii:
$B$ .
EXM.3.AHL.TZ0.5e.iv:
$D$ .
17M.1.AHL.TZ2.H_11b:
Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}$ .
18N.1.AHL.TZ0.H_11b.iii:
Hence, or otherwise, show that S = $\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)$ .
SPM.2.AHL.TZ0.12a:
Show that ${\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}$ .
18N.1.AHL.TZ0.H_11b.ii:
By writing $\frac{\pi }{{12}}$ as $\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)$ , find the value of cos $\frac{\pi }{{12}}$ in the form $\frac{{\sqrt a + \sqrt b }}{c}$ , where $a$ , $b$ and $c$ are integers to be determined.
16N.1.AHL.TZ0.H_13a:
Find the value of $\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}$ .
17M.1.AHL.TZ2.H_11c.ii:
Hence find the cube roots of $z$ in modulus-argument form.
19N.1.AHL.TZ0.H_4:
$A$ and $B$ are acute angles such that ${\text{cos}}\,A = \frac{2}{3}$ and ${\text{sin}}\,B = \frac{1}{3}$ .

Show that ${\text{cos}}\left( {2A + B} \right) = - \frac{{2\sqrt 2 }}{{27}} - \frac{{4\sqrt 5 }}{{27}}$ .
17M.1.AHL.TZ2.H_11a:
Solve $2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ$ .

Hide related questions

Topic 3— Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

View options