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.
Pythagoras' theorem: $a^2+b^2=c^2$a2+b2=c2, where 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.
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
$\sin23^\circ=\frac{43.6}{AC}$sin23°=43.6AC
$AC=\frac{43.6}{\sin23^\circ}$AC=43.6sin23°
$AC=111.59$AC=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
$\sin35.6^\circ=\frac{x}{37.2}$sin35.6°=x37.2
$x=37.2\times\sin35.6^\circ$x=37.2×sin35.6°
$x=21.65$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.