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Core 2: Trigonometry. Key formulae and ideas

Cosine Rule

 

 

 

 

 

 

 

 


There are 3 sides involved, a, b &c

There is an angle, A, which is opposite side a

 

 

OR

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Finding a side, x.

We can use the cosine rule because we KNOW TWO SIDES and the side we want is OPPOSITE the angles

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Finding a missing angle.

Notice how we KNOW THREE SIDES, so we can use the cosine-rule.

 

[the side opposite the angle]

 

 

 

 

 

 

 

 

Sine-Rule

 

 

 

 

 

 

 

 

 

 


 

Or

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Observe how the angles and sides are opposite each other. This means we can use the sine-rule.

 

 

 

 

 

 

 

 

 

 

 

 

 


Call missing angle x.

 

 

 

 

 


This is an interesting problem.

We want to find p, but we do not have an opposite side!

However, if we call the other missing angle x, then:

 

 

 

 

So to find p, we do:

 

 

 

 

 

 

 

Radians and degrees

 

Radians and degrees are two different scales for measuring angles.

There is a direct relationship between the two of them which makes it easy to convert between the two of them.

 

 

 

 

 

 


In other words

 

To convert from degrees to radians we:

 

Example:

 

 

 

To convert from radians to degrees we:

 

Examples:

 

Radians allow some properties of curved shapes to be worked out particularly quickly.

 

Arc and Sectors

 

 

 

 

 

 

 

 

 

 


This sector is a fraction of a circle.

The angle is measured in radians.

denotes the radius of the circle

 

 

 

 

 

 

KEY FORMULA FOR A SECTOR

 

 

 

 

 

 

 

Basic problem: Find area and perimeter

 

 

 

 

 

 

 

 

 

 

 

 


Angle already given in radians so:

 

 

More challenging problem:

 

A sector has an area of 5

 

The sector also has an arc length of 3.4

 

Find and

 

Solution

Turn the information into equations

 

 

Observe the trick:

So,

 

 

So,