Recap: basic trigonometric equations

Trigonometry, grade 11 (5 units) Lesson 2 of 18Learning

Recap: basic trigonometric equations

The general solution

A trigonometric equation has infinitely many solutions, since the functions repeat. The general solution writes all of them with a whole number $\displaystyle k$ :

$\displaystyle {\sin x}={a}$ (when $\displaystyle -1\le a\le1$ ): two angles at the same height on the unit circle, $\displaystyle \alpha$ and $\displaystyle \pi-\alpha$ .

\displaystyle x=\alpha+2\pi k\quad\text{or}\quad x=\pi-\alpha+2\pi k

$\displaystyle {\cos x}={a}$ : two angles mirrored in the $\displaystyle x$ -axis, $\displaystyle \alpha$ and $\displaystyle -\alpha$ .

\displaystyle x

\displaystyle =

\displaystyle \pm\alpha+2\pi k

$\displaystyle {\tan x}={a}$ : one angle in every half turn.

\displaystyle x

\displaystyle =

\displaystyle \alpha+\pi k

Here $\displaystyle \alpha$ is one angle with the value $\displaystyle a$ , and $\displaystyle k$ is any whole number.

\displaystyle {\sin f(x)}={\sin g(x)}

The same idea works when both sides are sines of expressions:

$\displaystyle {\sin f(x)}={\sin g(x)}$ : $\displaystyle {f(x)}={g(x)+2\pi k}$ or $\displaystyle {f(x)}={\pi-g(x)+2\pi k}$ .
$\displaystyle {\cos f(x)}={\cos g(x)}$ : $\displaystyle {f(x)}={\pm g(x)+2\pi k}$ .
$\displaystyle {\tan f(x)}={\tan g(x)}$ : $\displaystyle {f(x)}={g(x)+\pi k}$ .

An example: $\displaystyle {\cos3x}={\cos x}$ .

$\displaystyle {3x}={x+2\pi k}$ or $\displaystyle {3x}={-x+2\pi k}$ .
$\displaystyle {x}={\pi k}$ or $\displaystyle {x}={\frac{\pi k}{2}}$ .
Every multiple of $\displaystyle \pi$ is also a multiple of $\displaystyle \frac{\pi}{2}$ , so the solution is $\displaystyle {x}={\frac{\pi k}{2}}$ .

A multiple angle inside the sine

When the function holds $\displaystyle 2x$ , solve for $\displaystyle 2x$ first and only then divide – $\displaystyle 2\pi k$ too:

$\displaystyle {\sin2x}={\frac{\sqrt{3}}{2}}$

$\displaystyle {2x}={\frac{\pi}{3}+2\pi k}$ or $\displaystyle {2x}={\frac{2\pi}{3}+2\pi k}$ .
$\displaystyle {x}={\frac{\pi}{6}+\pi k}$ or $\displaystyle {x}={\frac{\pi}{3}+\pi k}$ .

After dividing, the period shrinks to $\displaystyle \pi$ : every full turn holds four solutions.

Degrees, and an inverse function

In degrees, $\displaystyle 360^\circ$ takes the place of $\displaystyle 2\pi$ and $\displaystyle 180^\circ$ of $\displaystyle \pi$ : the solution of $\displaystyle {\sin x}={\frac{1}{2}}$ is $\displaystyle {x}={30^\circ+360^\circ k}$ or $\displaystyle {x}={150^\circ+360^\circ k}$ .
When the value is not from the table, the angle $\displaystyle \alpha$ is written with the inverse function. The solution of $\displaystyle {\sin x}={\frac{1}{3}}$ : $\displaystyle {x}={\sin^{-1}\frac{1}{3}+2\pi k}$ or $\displaystyle {x}={\pi-\sin^{-1}\frac{1}{3}+2\pi k}$ .
The number comes from a calculator (in radian mode): $\displaystyle \sin^{-1}\frac{1}{3}\approx0.34$ .

Exercise 1

Solve.

Exercise 2

Solve.

Exercise 3

Solve.

Exercise 4

Solve.

Exercise 5

Solve in degrees.

Exercise 6

Solve. Write the solution with $\displaystyle \cos^{-1}$ .

Formulas: trigonometry, grade 11 (5 units)

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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
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^{2}$