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7 Further trigonometry
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10&10a

7.08 Applications

Lesson
Worksheet
Practice
Lesson

Introduction

Consider A,B,C as the vertices of a triangle and a,b,c as the side lengths opposite to each angle respectively. If this triangle is a non-right-angled triangle then we can use the sine rule, the cosine rule or the area rule to find unknown dimensions or the area. If the triangle is right-angled then we can use Pythagoras' theorem and the trigonometric ratios to find unknown dimensions.

Applications

We are now going to use these rules to find these unknown quantities in real world contexts. A good way to begin any question involving a triangle is to label the angles and their corresponding sides using the letters from the formula above. After that is done correctly we can use the rule that is most convenient.

Examples

Example 1

Find the length of the unknown side, x, in the given trapezium. Round your answer to two decimal places.

A trapezium B A C D with side lengths B A equal to 12, A C equal to 8, C D equal to 8, and B D equal to x.
Worked Solution
Create a strategy
A trapezium B A C D divided into a rectangle and right angled triangle. Ask your teacher for more information.

Divide the shape into a right angled triangle and a rectangle and use Pythagoras' theorem.

Apply the idea

From the right angled triangle, we see that a=8, b=12-6=6, and c=x.

\displaystyle c^2\displaystyle =\displaystyle a^2+b^2Write Pythagoras' theorem
\displaystyle x^2\displaystyle =\displaystyle 8^2+6^2Subsitute the values
\displaystyle =\displaystyle 100Evaluate the right side
\displaystyle \sqrt{x^2}\displaystyle =\displaystyle \sqrt{100}Take the square root of both sides
\displaystyle x\displaystyle =\displaystyle 10Evaluate and round

Example 2

Beth observes a tower at an angle of elevation of 12\degree. The tower is perpendicular to the ground.

Walking 75 m towards the tower, she finds that the angle of elevation increases to 34\degree.

The image shows two angles of elevation of 12 and 34 degrees to the top of a tower. Ask your teacher for more information.
a

Calculate the angle \angle ADB.

Worked Solution
Create a strategy

Use the exterior angle of a triangle rule.

Apply the idea
\displaystyle \angle ADB+12 \degree\displaystyle =\displaystyle 34\degreeUse the exterior angle of a triangle rule
\displaystyle \angle ADB\displaystyle =\displaystyle 22 \degreeSubtract 12 from both sides
b

Find the length of the side a correct to two decimal places.

Worked Solution
Create a strategy

Use the sine rule.

Apply the idea

In \triangle ABD we now have two angle side pairs to work with so we can use the sine rule.

\displaystyle \dfrac{a}{\sin 12\degree}\displaystyle =\displaystyle \dfrac{67}{\sin 22\degree}Use the sine rule
\displaystyle a\displaystyle =\displaystyle \dfrac{67\sin 12\degree}{\sin 22\degree}Multiply both sides by \sin 12\degree
\displaystyle \approx\displaystyle 37.19Evaluate and round
c

Evaluate the height h, of the tower. Round your answer to one decimal place.

Worked Solution
Create a strategy

Use the sine ratio.

Apply the idea

In \triangle BCD we now know the hypotenuse, one angle and need to find its opposite side. So we can use the sine ratio.

\displaystyle \sin \theta\displaystyle =\displaystyle \dfrac{\text{Opposite}}{\text{Hypotenuse}}Use the sine ratio
\displaystyle \sin 34\degree\displaystyle =\displaystyle \dfrac{h}{37.19}Substitute the values
\displaystyle h\displaystyle =\displaystyle 37.19\sin 34\degreeMultiply both sides by 37.19
\displaystyle \approx\displaystyle 20.8Evaluate and round

The height of the tower is 20.8\text{ m}.

Idea summary

We can use the following formulas for non-right angled triangles:

Sine rule\dfrac{\sin A}{a}=\dfrac{\sin B}{b}
Cosine rulec^2=a^2+b^2-2ab\cos C
Area rule\text{Area}=\dfrac{1}{2}ab\sin C

For right-angled triangles we can use the trigonometric ratios and Pythagoras' theorem:

Sine ratio\sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}}
Cosine ratio\cos \theta =\dfrac{\text{Adjacent}}{\text{Hypotenuse}}
Tangent ratio\tan \theta =\dfrac{\text{Opposite}}{\text{Adjacent}}
Pythagoras' theoremc^2=a^2+b^2

Outcomes

ACMMG273 (10a)

Establish the sine, cosine and area rules for any triangle and solve related problems

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