DP Mathematics SL Questionbank

• Syllabus

3.3

Sub sections and their related questions

The Pythagorean identity ${\cos ^2}\theta + {\sin ^2}\theta = 1$ .

• 08N.1.sl.TZ0.7a: Show that $f(x) = \sin x$ .
• 08M.1.sl.TZ1.2b: Find an expression for $\cos 140^\circ$ .
• 12M.1.sl.TZ1.7a: Show that $f(x)$ can be expressed as $1 + \sin 2x$ .
• 12M.1.sl.TZ1.7b: The graph of f is shown below for $0 \le x \le 2\pi$ . Let $g(x) = 1 + \cos x$ . On the...
• 12M.1.sl.TZ1.7c: The graph of g can be obtained from the graph of f under a horizontal stretch of scale factor p...
• 09N.1.sl.TZ0.6: Solve $\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x$ , for $0 \le x \le 2\pi$ .
• 15M.1.sl.TZ1.5a: find the value of $\cos x;$
• 16M.1.sl.TZ2.6a: Write $h(x)$ in the form $a\sin (bx)$, where $a,{\text{ }}b \in \mathbb{Z}$.
• 16M.1.sl.TZ2.6b: Hence find the range of $h$.
• 16N.1.sl.TZ0.2b: Find $\cos 2\theta$.
• 17M.1.sl.TZ2.7: Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) =  - 1$, for...
• 17N.1.sl.TZ0.6: Let $f(x) = 15 - {x^2}$, for $x \in \mathbb{R}$. The following diagram shows part of the...
• 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
• 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
• 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
• 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.

Double angle identities for sine and cosine.

• 08N.1.sl.TZ0.7b: Let $\sin x = \frac{2}{3}$ . Show that $f(2x) = - \frac{{4\sqrt 5 }}{9}$ .
• 08M.1.sl.TZ2.4a: Given that $\cos A = \frac{1}{3}$ and $0 \le A \le \frac{\pi }{2}$ , find $\cos 2A$ .
• 10N.1.sl.TZ0.5a: Show that $4 - \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin \theta + 3$ .
• 10N.1.sl.TZ0.5b: Hence, solve the equation $4 - \cos 2\theta + 5\sin \theta = 0$ for $0 \le \theta \le 2\pi$ .
• 10M.1.sl.TZ1.4a: Write down the value of $\tan \theta$ .
• 10M.1.sl.TZ1.4b(i) and (ii): Find the value of (i)     $\sin 2\theta$ ; (ii)    $\cos 2\theta$ .
• 10M.1.sl.TZ2.4a: Find $f\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4b: Find $(g \circ f)\left( {\frac{\pi }{2}} \right)$ .
• 10M.1.sl.TZ2.4c: Given that $(g \circ f)(x)$ can be written as $\cos (kx)$ , find the value of k,...
• 09N.1.sl.TZ0.6: Solve $\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x$ , for $0 \le x \le 2\pi$ .
• 09M.1.sl.TZ1.8b: Let the area of the rectangle be A. Show that $A = 18\sin 2\theta$ .
• SPNone.1.sl.TZ0.6a: Find the value of a and of b .
• 11N.1.sl.TZ0.6a: Find $\cos \theta$ .
• 11N.1.sl.TZ0.6b: Find $\tan 2\theta$ .
• 11M.2.sl.TZ2.10a: Show that the area of the window is given by $y = 4\sin \theta + 2\sin 2\theta$ .
• 11M.2.sl.TZ2.10b: Zoe wants a window to have an area of $5{\text{ }}{{\text{m}}^2}$. Find the two possible values...
• 11M.2.sl.TZ2.10c: John wants two windows which have the same area A but different values of $\theta$ . Find all...
• 14M.1.sl.TZ2.1b: Find $\cos 2A$.
• 15M.1.sl.TZ1.5b: find the value of $\cos 2x.$
• 16M.1.sl.TZ2.6a: Write $h(x)$ in the form $a\sin (bx)$, where $a,{\text{ }}b \in \mathbb{Z}$.
• 16M.1.sl.TZ2.6b: Hence find the range of $h$.
• 16N.1.sl.TZ0.2b: Find $\cos 2\theta$.
• 17M.1.sl.TZ2.7: Solve ${\log _2}(2\sin x) + {\log _2}(\cos x) =  - 1$, for...
• 17N.1.sl.TZ0.6: Let $f(x) = 15 - {x^2}$, for $x \in \mathbb{R}$. The following diagram shows part of the...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.10a.i: Find an expression for r in terms of θ.
• 18M.1.sl.TZ1.10a.ii: Find the possible values of r.
• 18M.1.sl.TZ1.10b: Show that the sum of the infinite sequence...
• 18M.1.sl.TZ1.10c: Find the values of θ which give the greatest value of the sum.

Relationship between trigonometric ratios.

• 12N.1.sl.TZ0.5a: Let $\sin {100^ \circ } = m$. Find an expression for $\cos {100^ \circ }$ in terms of m.
• 12N.1.sl.TZ0.5b: Let $\sin {100^ \circ } = m$ . Find an expression for $\tan {100^ \circ }$ in terms of m.
• 12N.1.sl.TZ0.5c: Let $\sin {100^ \circ } = m$. Find an expression for $\sin {200^ \circ }$ in terms of m.
• 08N.1.sl.TZ0.7b: Let $\sin x = \frac{2}{3}$ . Show that $f(2x) = - \frac{{4\sqrt 5 }}{9}$ .
• 08M.1.sl.TZ2.4b: Given that $\sin B = \frac{2}{3}$  and $\frac{\pi }{2} \le B \le \pi$ , find $\cos B$ .
• 12M.1.sl.TZ1.7a: Show that $f(x)$ can be expressed as $1 + \sin 2x$ .
• 12M.1.sl.TZ1.7b: The graph of f is shown below for $0 \le x \le 2\pi$ . Let $g(x) = 1 + \cos x$ . On the...
• 12M.1.sl.TZ1.7c: The graph of g can be obtained from the graph of f under a horizontal stretch of scale factor p...
• 09M.1.sl.TZ1.9c: (i)     Find ${\rm{sinR}}\widehat {\rm{P}}{\rm{Q}}$ . (ii)    Hence, find the area of triangle...
• 11N.1.sl.TZ0.6a: Find $\cos \theta$ .
• 11N.1.sl.TZ0.6b: Find $\tan 2\theta$ .
• 14M.1.sl.TZ2.1a: Show that $\cos A = \frac{{12}}{{13}}$.
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...