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 $c$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
$\sin23^\circ$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 |
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.
Find the length of the unknown side, $x$x, in the given trapezoid.
Give your answer correct to two decimal places.
Solve problems involving the measures of sides and angles in right triangles in real life applications, using the primary trigonometric ratios and the Pythagorean theorem.