Basic Trigonometry (Edexcel IGCSE Further Pure Maths)

Right-Angled Trigonometry

What is right-angled trigonometry?

• Right-angled trigonometry is the study of right-angled triangles

  • In particular the relationships between their side lengths and angles

• Right-angled trigonometry includes two main components

  • The Pythagorean theorem

  • SOHCAHTOA

• You should already be familiar with right-angled trigonometry from your regular IGCSE Mathematics course

What is the Pythagorean theorem?

• The Pythagorean theorem (or Pythagoras’ theorem ) connects the side lengths in a right-angled triangle

  • It only works for right-angled triangles!

• It says that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

  • The hypotenuse is always the longest side in a right-angled triangle

    • It will always be opposite the right angle

• If we label the hypotenuse c, and label the other two sides a and b, then

  • 

    • This is not on the exam formula sheet, so you need to remember it

• This lets us find one side length if we know the other two side lengths

  • To find the hypotenuse

    • 

  • To find one of the other two sides

    •   or  

• Pythagoras' theorem can also be used to prove that a triangle is (or isn't) right-angled

  • If    is true for three side lengths , and 

    • then the triangle is right-angled

  • If    is not true for three side lengths , and 

    • then the triangle is not right-angled

What is SOHCAHTOA?

• SOHCAHTOA is a way to help remember the sine, cosine and tangent formulae for right-angled triangles

  • These formulae only work for right-angled triangles!

• To use the formulae you must first label the sides of a right-angled triangle in relation to a chosen angle

  • The hypotenuse, H, is the longest side in a right-angled triangle

    • It will always be opposite the right angle

  • If we label one of the other angles θ,

    • the side opposite θ will be labelled opposite, O

    • and the side next to θ will be labelled adjacent, A

• The SOHCAHTOA formulae are

  •   ('SOH')

  •   ('CAH')

  •   ('TOA')

• These are not on the exam formula sheet, you need to remember them

• You can use SOHCAHTOA to find

  • an angle, if you know two side lengths

  • a side length, if you know an angle and another side length

• Start by choosing the correct formula

  • It needs to include two things you know as well as the thing you want to know

• Then substitute the values you know into the formula

  • and solve for the missing value

    • If finding an angle, you'll need to use ,  or  on your calculator

    • Or use your knowledge of exact trig values where possible

3D Problems

How does Pythagoras work in 3D?

• 3D shapes can often be broken down into several 2D shapes

• With Pythagoras’ Theorem you will be specifically looking for right-angled triangles

  • You need right-angled triangles with two known sides and one unknown side

  • Look for perpendicular lines to help you spot right-angled triangles

• There is a 3D version of the Pythagorean theorem formula:

  • 

    • is the length of the line segment

    • is the '-direction distance' between the endpoints of the line segment

    • is the '-direction distance'

    • is the '-direction distance' 

• However it is usually easier to break a 3D problem down into two or more 2D problems

How does SOHCAHTOA work in 3D?

• Again look for right-angled triangles including a missing angle or side

  • It may take more than one right-angles triangle to get to the answer

• The angle you are working with can be awkward in 3D

  • The angle between a line and a plane is not always obvious

  • If unsure choose a point on the line and draw a new line to the plane

    • This should create a right-angled triangle