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Trigonometry

Here are some tools to help you with
understanding trigonometry

What are the 3 main types of triangle?

What is Pythagoras's Theorem?

What is the SOHCAHTOA Relationship?

What are sine, cosine and tangents?

The 'Sine Rule'

The 'Cosine Rule'

Knowing when to use the sine and cosine rules

This section is a work in progress. Come back again soon.

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g) Arithmetic → h) Angle → i) Trigonometry → j) Algebra → k) Geometry

What are the 3 main types of triangle?

There are 3 types of triangle. They are shown below. The type of triangle and what you know of it will help determine which of the trigonometric rules you will need to apply to solve your problem.

β (beta) is the lower-case Greek symbol for the letter b and γ (gamma) is the lower-case Greek symbol for the letter c.

Mathematicians sometimes use these letters to avoid making mistakes telling the angles apart from the sides (as opposed to just using capitals).

What is Pythagoras's Theorem?

Named after the Greek philosopher Pythagoras (who didn't create it but popularised it), this helps us find a missing side length in a right-angled triangle as long as two other side lengths are known.

BEWARE - This ONLY works for RIGHT-ANGLED TRIANGLES!

What is the SOHCAHTOA Relationship?

The SOHCAHTOA relationship is a MNENONIC that is useful for quickly rearranging a formula to find an unknown.

BEWARE - This ONLY works for RIGHT-ANGLED TRIANGLES!

Work in progress - The angles do not draw correctly - Yet!!!

What are sine, cosine and tangents?

'Sine' and 'Cosine' values can be thought of as the coordinates of a point that sits on a circle in relation to the centre of the circle. 'Sine' represents the X coordinate and 'Cosine' represents the Y coordinate. They are both unitless properties (they don't carry units).

A 'tangent' can be thought of as the slope of the projected line as it intersects the circle. Since it is always the sine divided by the cosine (i.e. a fraction or division) it too is unitless. If you drew a line from the circle at the end of the X axis until it 'hits' the projected line, that would be the tangent.

All three properties are essential to understanding trigonometry and the 'SOH-CAH-TOA' relationship.

The 'Sine Rule'

Use the sine rule when:

You have a known angle–opposite side pair; and
you know either:
- another angle and need a side (AAS or ASA), or
- another side and need an angle (SSA).

In summary:

The sine rule is used when you know or can find a matching side–angle pair.

ACRONYMS
AAS = Angle-Angle-Side
SSA = Side-Side-Angle
SAS=Side-Angle-Side
SSS=Side-Side-Side

The 'Cosine Rule'

Use the cosine rule when:

You do not have a pair of an angle and its opposite side; or
You know two sides and the included angle (SAS)
and need the third side; or
You know all three sides (SSS) and need to find an angle.

In summary:

The cosine rule is used when the triangle is not suitable for the sine rule — typically when you’re missing an opposite pair.

ACRONYMS
AAS = Angle-Angle-Side
SSA = Side-Side-Angle
SAS=Side-Angle-Side
SSS=Side-Side-Side

Knowing when to use the sine and cosine rules

Where triangles are concerned, if you know 2 values (angles or side lengths), then you can find a third. There are two 'rules' (i.e. expressions) for doing this; the 'Sine rule' and the 'Cosine rule'. Depending upon what you know (or have been 'given'), then this will determine which of the rules you will need to use.

This tool is a work in progress. Beware that it is quite buggy at present.

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