3.12 Various applications

Using trigonometry to solve problems in the real world involves not only identifying the relevant triangles, but also knowing which trigonometric tools to use to get the information we want. Being able to choose the correct tool is an important skill for solving any problem. Lets have another look at the tools we can now use:

• Pythagoras' theorem
• Trigonometric ratios
• The sine rule
• The cosine rule
• The area rule

The last three of these can be used for any triangle, however the first two in the list depend on the triangle having a right-angle.

Pythagoras' Theorem and trigonometric ratios

Remember, these tools are only able to be used in triangles which are right-angled. 

Relationships in right-angled triangles

For the triangle shown, the following equations are always true:

$a^2+b^2=c^2$a2+b2=c2

and

$\sin\theta=\frac{a}{c},\cos\theta=\frac{b}{c},\tan\theta=\frac{a}{b}$sinθ=ac​,cosθ=bc​,tanθ=ab​

The following table describes what which right-angled triangle rules you can use and how, given the information you already know. Notice in these cases that we always know the size of at least one angle, which is the right-angle.

The sine and cosine rules

These rules describe the relationship between different sides and angles of a triangle, whether the triangle has a right-angle or not.

The sine and cosine rules

For the triangle shown, the following equations are always true:

$\frac{\sin A}{a}$sinAa​$=$=$\frac{\sin B}{b}=\frac{\sin C}{c}$sinBb​=sinCc​ (the sine rule)

and

$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC (the cosine rule)

As we've noticed in previous chapters, each rule can be used in different situations because they rely on different information to solve for an angle size or side length. When you see a triangle and want to solve for the information you don't know, the first step is to figure out what you do know about the triangle. Then, choose the rule which is able to use that information.

The following table describes what rules you can use and how, given the information you know about a triangle.

The area rule

Like the sine and cosine rules, the area rule can be used for any triangle. 

The area rule

For the triangle shown, the following equation is always true:

 

$Area$Area$=$=$\frac{1}{2}ab\cos C$12​abcosC

Since this is the only rule so far that relates the side lengths and angles of a triangle to its area, this rule will be helpful whenever the area of a triangle is unknown.

Practice questions

QUESTION 1

QUESTION 2

QUESTION 3