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

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.

What is the value of $x$x? Give your answer correct to two decimal places.

Using the value of $x$x you got from part (a), find the value of $y$y correct to two decimal places.

Question 3

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