Trigonometric Equations (Edexcel IGCSE Further Maths): Exam Questions

Solve, in degrees to 1 decimal place, for $0 \text{less} - \text{than} or \text{slanted} \text{equal} to \theta < 180$

$2\mathrm{cos}(2θ+30)^{\circ}+\mathrm{tan}(2θ+30)^{\circ}=0$

Using formulae , show that

(i) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

(ii) $\mathrm{cos}2A=2\mathrm{cos}^{2}A-1$

$\mathrm{f} ( \\theta ) = \frac{2 \tan \\theta }{1 + \tan ^{2} \\theta }$

Show that $f(θ)=\mathrm{sin}2θ$

Solve, in radians to 3 significant figures, for negative pi over 2 less-than or slanted equal to x less-than or slanted equal to pi over 2, the equation

$5 \tan ( x + \frac{\pi }{6} ) = [ 1 + \tan ^{2} ( x + \frac{\pi }{6} ) ] [ 1 - 2 \cos ^{2} ( x + \frac{\pi }{6} ) ]$

Using calculus, find the exact value of

$\int_{0}^{ \frac{\pi }{2} } ( \frac{4 \tan \\theta }{1 + \tan ^{2} \\theta } - \cos 5 \\theta + 2 )   \mathrm{d} \\theta$

Using a formula , show that

$\mathrm{tan}2A=\frac{2\mathrm{tan}A}{1-\mathrm{tan}^{2}A}$

Hence, solve the equation

$\mathrm{tan}A^{\circ}-\mathrm{tan}2A^{\circ}=0$ for 0 less-than or slanted equal to A less-than or slanted equal to 180

Using a formula given on page 2, solve, giving your solutions as exact values

$\cos ( x - \frac{\pi }{6} ) = \sin x$ for negative pi less-than or slanted equal to x less-than or slanted equal to 2 pi

Using formulae , show that

(i) $\mathrm{cos}2A=2\mathrm{cos}^{2}A-1$

(ii) $\mathrm{sin}2A=2\mathrm{sin}A\mathrm{cos}A$

Show that $\mathrm{cos}^{3}A=\frac{\mathrm{cos}3A+3\mathrm{cos}A}{4}$

Hence, or otherwise, solve, giving exact values in terms of $π$

$8 \text{cos} ^{3} ( \frac{ \theta }{2} ) - 6 \text{cos} ( \frac{ \theta }{2} ) - 1 = 0$ for 0 less-than or slanted equal to theta less-than or slanted equal to 2 pi

Use algebraic integration to find the exact value of

integral _ 0 ^ pi / 6 end ^ ( 4 cos ^3 theta - sin 2 theta ) d theta

Using the formulae , show that

(i) $\mathrm{sin}2θ=2\mathrm{sin}θ\mathrm{cos}θ$

(ii) $\mathrm{cos}2θ=2\mathrm{cos}^{2}θ-1$

Given that $θ\neq(90^{\circ}+180^{\circ}n)$ where $n\inℤ$

use the results from part (a) to show that $\mathrm{sin}2θ-\mathrm{tan}θ$ can be written as $\mathrm{tan}θ\mathrm{cos}2θ$

Solve for $0<x<360$

$\mathrm{sin}2x^{\circ}-\mathrm{tan}x^{\circ}=0$

Using the formulae , show that

(i) $\mathrm{cos}^{2}A=\frac{\mathrm{cos}2A+1}{2}$

(ii) $\mathrm{sin}^{2}A=\frac{1-\mathrm{cos}2A}{2}$

Show that

$(2\mathrm{sin}x-\mathrm{cos}x)(\mathrm{sin}x-3\mathrm{cos}x)=\frac{1}{2}(\mathrm{cos}2x-7\mathrm{sin}2x+5)$

$y=(2\mathrm{sin}x-\mathrm{cos}x)(\mathrm{sin}x-3\mathrm{cos}x)$

Solve, for 0 degree less-than or slanted equal to x less-than or slanted equal to 180 degree the equation, $\frac{dy}{dx}=0$

Give your answers to the nearest whole number.