Trigonometry

Ontario 10 Academic (MPM2D)

Triangle problems

Lesson

So far we have found unknown side lengths using Pythagoras' theorem and then looked at 3 special ratios that we can use to find unknown sides or angles in right-angled triangles.

Right-angled triangles

Pythagoras' theorem: $a^2+b^2=c^2$`a`2+`b`2=`c`2, where c is the hypotenuse

$\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$`s``i``n``θ`=Opposite Hypotenuse = $\frac{O}{H}$`O``H`

$\cos\theta=\frac{\text{Adjacent }}{\text{Hypotenuse }}$`c``o``s``θ`=Adjacent Hypotenuse = $\frac{A}{H}$`A``H`

$\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$`t``a``n``θ`=Opposite Adjacent =$\frac{O}{A}$`O``A`

Problem solving in trigonometry can be in finding unknowns like we have already been doing, using trigonometry in real world applications or in solving geometrical problems like these.

Find $x$`x` in the following geometrical diagram,

**Think**: In order to find $x$`x`, I will need to identify some other measurements along the way. My problem solving strategy will be

1. Find length $AC$`A``C` using trig ratio sine

2. Find length $ED$`E``D`, $\frac{AC}{3}$`A``C`3

3. Find length $x$`x`, using trig ratio sine

**Do**:

1. Find length $AC$`A``C` using trig ratio sine

$\sin23^\circ=\frac{43.6}{AC}$`s``i``n`23°=43.6`A``C`

$AC=\frac{43.6}{\sin23^\circ}$`A``C`=43.6`s``i``n`23°

$AC=111.59$`A``C`=111.59

2. Find length $ED$`E``D`, $\frac{AC}{3}$`A``C`3

$ED=\frac{111.59}{3}$`E``D`=111.593

$ED=37.2$`E``D`=37.2

3. Find length $x$`x`, using trig ratio sine

$\sin35.6^\circ=\frac{x}{37.2}$`s``i``n`35.6°=`x`37.2

$x=37.2\times\sin35.6^\circ$`x`=37.2×`s``i``n`35.6°

$x=21.65$`x`=21.65

Find the length of the unknown side in this right-angled triangle, expressing your answer as a decimal approximation to two decimal places.

Find the value of $f$`f`, correct to two decimal places.

Find the value of $x$`x` to the nearest degree.

Find the value of $x$`x`, the side length of the parallelogram, to the nearest centimetre.

Consider the given figure.

Find the unknown angle $x$

`x`, correct to two decimal places.Find $y$

`y`, correct to two decimal places.Find $z$

`z`correct to two decimal places.

Determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem