Ontario 10 Academic (MPM2D)
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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 trapezoid.

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

 

Outcomes

10D.T2.03

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.

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