Equations with identities

Trigonometry, grade 11 (5 units) Lesson 9 of 18Learning

Equations with identities

The way to a solution

A complicated trigonometric equation is solved in two stages:

With identities, bring it to one function of one angle – $\displaystyle \sin x$ , $\displaystyle \cos x$ or $\displaystyle \tan x$ – or to a product equal to $\displaystyle 0$ .
Solve basic equations ( $\displaystyle {\sin x}={a}$ and the like) and write the general solution.

A quadratic equation in

\displaystyle \sin x

In the example below $\displaystyle \cos^2x$ is replaced by $\displaystyle 1-\sin^2x$ , so that one function is left.

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

$\displaystyle {2(1-\sin^2x)+3\sin x-3}={0}$ , that is $\displaystyle {2\sin^2x-3\sin x+1}={0}$ .
This is a quadratic equation in $\displaystyle \sin x$ : $\displaystyle {\sin x}={1}$ or $\displaystyle {\sin x}={\frac{1}{2}}$ .
$\displaystyle {x}={\frac{\pi}{2}+2\pi k}$ or $\displaystyle {x}={\frac{\pi}{6}+2\pi k}$ or $\displaystyle {x}={\frac{5\pi}{6}+2\pi k}$ .

A root of the quadratic outside $\displaystyle -1$ to $\displaystyle 1$ (say $\displaystyle {\sin x}={2}$ ) gives no solutions.

The double angle and factorising

$\displaystyle {\sin2x}={\cos x}$

$\displaystyle {2\sin x\cos x-\cos x}={0}$ , that is $\displaystyle {\cos x(2\sin x-1)}={0}$ .
$\displaystyle {\cos x}={0}$ or $\displaystyle {\sin x}={\frac{1}{2}}$ .
$\displaystyle {x}={\frac{\pi}{2}+\pi k}$ or $\displaystyle {x}={\frac{\pi}{6}+2\pi k}$ or $\displaystyle {x}={\frac{5\pi}{6}+2\pi k}$ .

Careful: dividing $\displaystyle {2\sin x\cos x}={\cos x}$ by $\displaystyle \cos x$ loses the solutions where $\displaystyle {\cos x}={0}$ . Take out the common factor – do not divide.

A homogeneous equation

In a homogeneous equation every term has the same degree in $\displaystyle \sin x$ and $\displaystyle \cos x$ together, as in $\displaystyle {2\sin^2x-3\sin x\cos x+\cos^2x}={0}$ . Here dividing by $\displaystyle \cos^2x$ is allowed: were $\displaystyle {\cos x}={0}$ , the equation would give $\displaystyle {2\sin^2x}={0}$ , but $\displaystyle \sin x$ and $\displaystyle \cos x$ are never 0 together.

$\displaystyle {2\tan^2x-3\tan x+1}={0}$ .
$\displaystyle {\tan x}={1}$ or $\displaystyle {\tan x}={\frac{1}{2}}$ .
$\displaystyle {x}={\frac{\pi}{4}+\pi k}$ or $\displaystyle {x}={\tan^{-1}\frac{1}{2}+\pi k}$ .

\displaystyle {a\sin x+b\cos x}={c}

Write the left side as one sine, as in the lesson on the sum formulas:

$\displaystyle {\sin x+\sqrt{3}\cos x}={1}$

$\displaystyle {2\sin\left(x+\frac{\pi}{3}\right)}={1}$ , that is $\displaystyle {\sin\left(x+\frac{\pi}{3}\right)}={\frac{1}{2}}$ .
$\displaystyle {x+\frac{\pi}{3}}={\frac{\pi}{6}+2\pi k}$ or $\displaystyle {x+\frac{\pi}{3}}={\frac{5\pi}{6}+2\pi k}$ .
$\displaystyle {x}={-\frac{\pi}{6}+2\pi k}$ or $\displaystyle {x}={\frac{\pi}{2}+2\pi k}$ .

Exercise 1

Solve.

Exercise 2

Solve.

Exercise 3

Solve. A hint: which form of $\displaystyle \cos2x$ leaves $\displaystyle \sin x$ alone?

Exercise 4

Solve.

Exercise 5

Solve.

Exercise 6

Solve.

Formulas: trigonometry, grade 11 (5 units)

Open as a page

Basic identities

The Pythagorean identity
$\displaystyle \sin^{2}\alpha+\cos^{2}\alpha$ $\displaystyle =$ $\displaystyle 1$
The tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$

The sum and the difference of angles

$\displaystyle \cos(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \cos\alpha\cdot\cos\beta\mp\sin\alpha\cdot\sin\beta$
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$
$\displaystyle \tan(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\cdot\tan\beta}$

The double angle

$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
$\displaystyle \cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha=2\cos^{2}\alpha-1=1-2\sin^{2}\alpha$
$\displaystyle \tan2\alpha$ $\displaystyle =$ $\displaystyle \frac{2\tan\alpha}{1-\tan^{2}\alpha}$
$\displaystyle \cos^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1+\cos2\alpha}{2}$
$\displaystyle \sin^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1-\cos2\alpha}{2}$

Sums and differences of sines and of cosines

$\displaystyle \sin\alpha+\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha-\beta}{2}\cdot\cos\frac{\alpha+\beta}{2}$
$\displaystyle \cos\alpha+\cos\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \cos\alpha-\cos\beta$ $\displaystyle =$ $\displaystyle -2\sin\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$

Equations – the general solution ($k$ a whole number)

$\displaystyle \sin x=\sin\alpha\iff x=\alpha+2\pi k\ \text{or}\ x=\pi-\alpha+2\pi k$
$\displaystyle \cos x=\cos\alpha\iff x=\pm\alpha+2\pi k$
$\displaystyle \tan x=\tan\alpha\iff x=\alpha+\pi k$

A triangle

The law of sines ( $\displaystyle R$ – the radius of the circumscribed circle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^{2}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}-2a\cdot b\cdot\cos\gamma$
The area of a triangle ( $\displaystyle \alpha$ – the angle between $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}b\cdot c\cdot\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ , $\displaystyle \gamma$ – the angles next to $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^{2}\cdot\sin\beta\cdot\sin\gamma}{2\cdot\sin(\beta+\gamma)}$

A circle ($\alpha$ in radians, $R$ – the radius)

The area of a disc
$\displaystyle S$ $\displaystyle =$ $\displaystyle \pi R^{2}$
The circumference of a circle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\pi R$
The length of an arc
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha\cdot R$
The area of a sector
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha\cdot R^{2}$

Solids ($S$ – the base area, $h$ – the height, $l$ – the slant height)

The volume of a prism and a cylinder
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral surface of a cone
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\cdot l$
The volume of a sphere
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^{3}$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^{2}$