# Solve problems involving two right-angled triangles

Lesson

The next step in our trigonometric problem solving adventure is to solve 2, 3 or more step problems.  What I mean is, solve problems where you have to solve other intermediate steps along the way.

The best way to learn these is to watch some examples, and then try the set of questions.

You'll need to remember these right-angled triangle rules:

Right-angled triangles

Pythagoras' theorem:  $a^2+b^2=c^2$a2+b2=c2, where $c$c is the hypotenuse

$\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$sinθ=Opposite Hypotenuse = $\frac{O}{H}$OH

$\cos\theta=\frac{\text{Adjacent }}{\text{Hypotenuse }}$cosθ=Adjacent Hypotenuse = $\frac{A}{H}$AH

$\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$tanθ=Opposite Adjacent =$\frac{O}{A}$OA

Angle of Elevation: the angle from the observer's horizontal line of sight looking UP at an object

Angle of Depression: the angle from the observer's horizontal line of sight looking DOWN at an object

Exact value triangles

#### Examples

##### Question 1

Consider the following diagram.

1. Find the length of $AD$AD, correct to 2 decimal places.

2. Find the length of $BD$BD, correct to 2 decimal places.

3. Hence, find the length of $AB$AB correct to 2 decimal places.

##### Question 2

Consider the following diagram.

1. Find $y$y, correct to two decimal places.

2. Find $w$w, correct to two decimal places.

3. Hence, find $x$x, correct to one decimal place.

### Outcomes

#### 10P.MT2.02

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