**Trigonometry**

5.6 The Sine Law

Sine law is used to solve two-dimensional problems. The sine law can be used given two sides and one angle opposite the given side (SSA) or two angles and any side (AAS or ASA).

The following equation represents the sine law:

5.7 The Cosine Law

5.8 Solving 3D Problems

Three dimensional problems involving triangles can be solved by using the sine law, cosine law or the Pythagorean theorem. Depending on the information given you can determine which law or formula to use. Don't forget to start yourself off with a sketch of the given information.

**Filsan Nur & Lidwyne Fonrose**

Example 1:

A triangular plot of land is enclosed by a fence. One side of the fence is 8.1m long with an opposite angle of 75 degrees. An adjacent side of the fence is 5.7m long with an opposite angle of theta.

Answer: 43 degrees

a) Make a sketch of the situation.

b) Determine angle theta to the nearest degree.

Note: Don't forget to set to calculators to Degree mode!

The Ambiguous Case

Problems like these can become confusing and one can become easily overwhelmed. Make sure that you break the question down into steps to find your final answer! :)

The cosine law is used if you are given two sides and an angle contained between those sides (SAS) or if the three side lengths are given.

The following equation represents of the cosine law:

Example 3:

In triangle ABC, a=11.5, b=8.3, and c=6.6. Calculate angle A to the nearest degree.

Answer: 100 degree

If you are given two angles and one side (ASA or AAS), the Law of Sines will nicely provide you with ONE solution for a missing side.

Unfortunately, this law has a problem when you're dealing with two sides and an angle and you want to find a missing angle. In this case, the law could provide you with one, two or no solutions.

How to Solve

1. Compare the values of a and b. If a>b, then there is only one triangle. Proceed to step 3.

2. When a<b, calculate bsinA

if a<bsinA - there are no triangles

if a=bsinA - one right triangle

if bsinA<a<b - 2 triangles, one acute, one obtuse

3. Use sine law to measure the <B

Don't forget if there are two triangles, the second obtuse can be found by 180 - <B!

Example 2:

In triangle ABC, side b is 3 cm long, side c is 5.5cm and <B is 30 degrees, Solve all sides and angles.

*a is always the side

OPPOSITE

to the angle given*

HINT:

Example 4

Determine the value of x to the nearest centimetre and theta to the nearest degree.

a)

An

s

w

e

r:

1

7

c

m