Trigonometric Equations (Edexcel AS Maths: Pure): Exam Questions

Solve the equation

$2{\sin }^{2}3x=1$

for $-90°\le x\le 90°$.

Express

$\left(x+1\right)\left(x-2\right)\left(x-3\right)$

in the form

$a{x}^3+b{x}^2+cx+d$

where $a$, $b$, $c$ and $d$ are constants to be found.

Hence solve the equation

${\tan }^{3}x-4{\tan }^{2}x+\tan x+6=0$

for $0°\le x\le 360°$.

Give your answers to 1 decimal place where necessary.

Express the equation

$2{\sin }^{2}x+3\cos x=0$

in the form

$a{\cos }^{2}x+b\cos x+c=0$

where $a$, $b$ and $c$ are constants to be found.

Hence solve the equation

$2{\sin }^{2}x+3\cos x=0$

for $-180°\le x\le 180°$.

Show that the equation

$4\cos \theta -1=2\sin \theta \tan \theta$

can be written in the form

$6{\cos }^2\theta -\cos \theta -2=0$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Hence solve, for $0⩽x⩽360°$, the equation

$4\cos 2x-1=2\sin 2x\tan 2x$

giving your answers to one decimal place.

Solve the equation

$2\sin \theta =3\cos \theta$

for $0°\le \theta \le 360°$.

Give your answers to 1 decimal place.

Given that angle $\theta$ is obtuse where

$\sin \theta =\frac{3}{4}$

use a non-calculator method to find the exact value of $\cos \theta$.

Given that angle $\theta$ is reflex where

$\cos \theta =\frac{1}{3}$

use a non-calculator method to find the exact value of $\tan \theta$.

Show that

$\frac{1-{\cos }^{2}x}{{\tan }^{2}x}\equiv {\cos }^{2}x$

Solve the equation

$2\sin x=\frac{1}{\sin x}$

for $0°\le x\le 360°$.

Solve the equation

$2{\sin }^2\theta =1+\cos \theta$

for $-180°\le \theta \le 180°$.

Solve the equation

$2\sin x\cos x=\cos x$

for $-180°\le x\le 180°$.

A seagull sits on the surface of the sea, moving up and down with the waves.

Its height, $h$ metres, above sea level in calm water is modelled by

$h=\frac{1}{2}\sin \left(180t\right)°$

where $t$ is the time in seconds after first being observed.

Sketch the graph of $h$ against $t$ for $0\le t\le 10$, showing the coordinates of the points of intersection with the $t$ axis.

Find the time at which the seagull is first observed to be 0.25 m above sea level in calm water and moving downwards.

Give your answer to 3 significant figures.

Solve the following equation, for $0°\le x\le 360°$

$2\sin x=\cos x$

Show that

$\left(1-{\sin }^{\mathit{2}}x\right){\tan }^{2}x\equiv {\sin }^{2}x$

$\mathrm{f}\left(x\right)=-3x^3+8x^2-9x+10,x\in \mathrm{ℝ}$

(i) Calculate $\mathrm{f}\left(2\right)$

(ii) Write $\mathrm{f}\left(x\right)$ as a product of two algebraic factors.

Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation

$-3y^6+8y^4-9y^2+10=0$

Deduce the number of real solutions, for $7\pi \le \theta <10\pi$, to the equation

$3{\tan }^3\theta -8{\tan }^2\theta +9\tan \theta -10=0$

Solve the equation

$\tan 2x=\frac{3}{\tan 2x}$

for $-180°\le x\le 180°$.

Solve the equation 

$3\sin 3\theta =4\cos 3\theta$

for $0°\le \theta \le 180°$.

Give your answers to 1 decimal place.

Solve the equation

$2\tan x-\sin x=0$

for $-180°\le x\le 180°$.

Show that $x=\frac{1}{2}$ is a solution to the equation

$8x^3-4x^2-6x+3=0$

Hence solve

$8{\cos }^{3}x-4{\cos }^{2}x-6\cos x+3=0$

for $0°\le x\le 360°$.

Solve the equation

$6{\cos }^2\left(2\theta \right)=5+\sin \left(2\theta \right)$

for $-180°\le \theta \le 180°$.

Give your answers to 1 decimal place where necessary.

Solve the equation

$3\sin \left(2x+30°\right)=\tan \left(2x+30°\right)$

for $-180°\le x\le 180°$.

Give your answers to 1 decimal place where necessary.

Show that

$\begin{array}{ccccc}\frac{1}{\cos \theta }+\tan \theta \equiv \frac{\cos \theta }{1-\sin \theta }& & \theta \ne \left(2n+1\right)90°& & n\in \mathrm{\mathbb{Z}}\end{array}$

Given that $\cos 2x\ne 0$

solve for $0<x<90°$

$\frac{1}{\cos 2x}+\tan 2x=3\cos 2x$

giving your answers to one decimal place.

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Show that the equation

$4\tan x=5\cos x$

can be written as

$5{\sin }^{2}x+4\sin x-5=0$

Hence solve, for $0<x\le 360°$

$4\tan x=5\cos x$

giving your answers to one decimal place.

Hence find the number of solutions of the equation

$4\tan 3x=5\cos 3x$

in the interval $0<x\le 1800°$, explaining briefly the reason for your answer.

In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Solve, for $0<\theta \le 450°$, the equation

$5{\cos }^2\theta =6\sin \theta$

giving your answers to one decimal place.

A student’s attempt to solve the question

“Solve, for $–90°<x<90°$, the equation $3\tan x–5\sin x=0$”

is set out below.

Identify two errors or omissions made by this student, giving a brief explanation of each.

The first four positive solutions, in order of size, of the equation

$\cos \left(5\alpha +40°\right)=\frac{3}{5}$

are ${\alpha }_1$ , ${\alpha }_2$, ${\alpha }_3$ and ${\alpha }_4$

Find, to the nearest degree, the value of ${\alpha }_4$

For the triangle in the diagram below, find the exact values of $\sin x$, $\cos x$ and $\tan x$.

Find all the values of $x$ in the interval $0°\le x\le 180°$  which satisfy

$6{\tan }^{3}2x-7{\tan }^{2}2x-\tan 2x+2=0$

giving your answers to 1 decimal place.