Recap: the law of sines and the law of cosines

Trigonometry, grade 11 (5 units) Lesson 13 of 18Learning

Recap: the law of sines and the law of cosines

The three tools

In triangle $\displaystyle ABC$ the side $\displaystyle a$ is opposite the angle $\displaystyle \alpha$ , the side $\displaystyle b$ opposite $\displaystyle \beta$ , and the side $\displaystyle c$ opposite $\displaystyle \gamma$ . $\displaystyle R$ is the radius of the circle circumscribing the triangle.

The law of sines – a ratio of a side to the sine of the angle opposite it:

\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R

The law of cosines – a side from the two other sides and the angle between them:

\displaystyle c^2

\displaystyle =

\displaystyle a^2+b^2-2ab\cos\gamma

The area – two sides and the angle between them:

\displaystyle S

\displaystyle =

\displaystyle \frac{1}{2}ab\sin\gamma

In the lessons on triangles the angles are in degrees, and every result is rounded to two decimal places, only at the end of the working (≈).

Which law to choose

Given	Law
two angles and a side	sines (the third angle: $\displaystyle 180^\circ$ minus the two)
two sides and the angle between them	cosines
three sides	cosines – to find an angle
two sides and an angle opposite one of them	sines – there may be two triangles

Two angles and a side

In triangle $\displaystyle ABC$ : $\displaystyle {BC}={8}$ , $\displaystyle {\beta}={40^\circ}$ , $\displaystyle {\gamma}={75^\circ}$ .
Find $\displaystyle AC$ and the radius of the circumscribed circle.

$\displaystyle {\alpha}={180^\circ-40^\circ-75^\circ}={65^\circ}$ .
$\displaystyle {\frac{AC}{\sin40^\circ}}={\frac{8}{\sin65^\circ}}$ , so $\displaystyle {AC}={\frac{8\sin40^\circ}{\sin65^\circ}\approx5.67}$ .
$\displaystyle {2R}={\frac{8}{\sin65^\circ}}$ , so $\displaystyle {R}={\frac{8}{2\sin65^\circ}\approx4.41}$ .

Two triangles

In triangle $\displaystyle ABC$ : $\displaystyle {BC}={7}$ , $\displaystyle {AC}={5}$ , $\displaystyle {\beta}={40^\circ}$ .
Find $\displaystyle \alpha$ .

$\displaystyle {\frac{7}{\sin\alpha}}={\frac{5}{\sin40^\circ}}$ , so $\displaystyle {\sin\alpha}={\frac{7\sin40^\circ}{5}\approx0.90}$ .
An acute angle and the obtuse angle that adds up with it to $\displaystyle 180^\circ$ have the same sine: $\displaystyle \alpha\approx64.15^\circ$ or $\displaystyle \alpha\approx115.85^\circ$ .
In both cases $\displaystyle \alpha+\beta<180^\circ$ , so there are two such triangles – in the drawing, $\displaystyle ABC$ and $\displaystyle DBC$ .

Exercise 1

In triangle $\displaystyle ABC$ : $\displaystyle {AB}={6}$ , $\displaystyle {AC}={9}$ , $\displaystyle {\alpha}={50^\circ}$ .

a

Find

\displaystyle BC

.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find the area of the triangle.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The sides of a triangle are $\displaystyle 5$ , $\displaystyle 6$ and $\displaystyle 7$ long.

a

Find the largest angle of the triangle. (Which side is it opposite?)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find the radius of the circle circumscribing the triangle.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

In triangle $\displaystyle ABC$ : $\displaystyle {AB}={10}$ , $\displaystyle {\alpha}={35^\circ}$ , $\displaystyle {\beta}={105^\circ}$ .
Find $\displaystyle BC$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

In triangle $\displaystyle ABC$ : $\displaystyle {BC}={6}$ , $\displaystyle {AC}={8}$ , $\displaystyle {\alpha}={30^\circ}$ .
Find both possible values of $\displaystyle \beta$ , and $\displaystyle AB$ in each of the triangles.

Acute

\displaystyle \beta

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle AB

(acute

\displaystyle \beta

)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Obtuse

\displaystyle \beta

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle AB

(obtuse

\displaystyle \beta

)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: trigonometry, grade 11 (5 units)

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