UK Secondary (7-11) Applications to Geometry
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:  $a^2+b^2=c^2$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

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.

#### Examples

##### Question 1

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 $AC$AC using trig ratio sine

2. Find length $ED$ED, $\frac{AC}{3}$AC3

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

Do:

1. Find length $AC$AC using trig ratio sine

 $\sin23^\circ$sin23° $=$= $\frac{43.6}{AC}$43.6AC​ $AC$AC $=$= $\frac{43.6}{\sin23^\circ}$43.6sin23°​ $AC$AC $=$= $111.59$111.59

2. Find length $ED$ED, $\frac{AC}{3}$AC3

$ED=\frac{111.59}{3}$ED=111.593

$ED=37.2$ED=37.2

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

 $\sin35.6^\circ$sin35.6° $=$= $\frac{x}{37.2}$x37.2​ $x$x $=$= $37.2\times\sin35.6^\circ$37.2×sin35.6° $x$x $=$= $21.65$21.65

##### Question 2

Consider the following diagram. 1. What is the value of $x$x? Give your answer correct to 2 decimal places.

2. Using the rounded value of $x$x, find the value of $y$y. Give your answer correct to 2 decimal places.

##### Question 3

Find the length of the unknown side, x, in the given trapezium.

Give your answer correct to $2$2 decimal places. 