Trigonometric Ratios/Pythagoras' Theorem

General Information
'Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions.' is a bullet point under Level 6 Mathematics and Statistics, Geometry and Measurement. 'Trigonometric ratios' here refers to the Sine Rule, which can be very useful in finding unknown sides of a triangle, no matter its type. The Sine Rule and Pythagoras' Theorem are neccesary knowledge for a geometric reasoning exam, and are usually assessed with other rules, especially in 'bearings' questions.

What you need to know
The Law of Sines basically states that in any triangle, the ratio between sin(an angle) and the side opposite it is equal to sin(another angle) and the side opposite that angle. Using the example of the picture to the right, sinA/a=sinB/b=sinC/c. This gives us a way to solve for a side or angle of a triangle, knowing the side/angle opposite it and any other side and the angle opposite that. Simply find the ratio as a number, and then use that number in sinx/(given side)=(ratio)

Pythagoras' Theorem states that in a right angle triangle, the sum of the squares of the two shorter side lengths equal the square of the longer side's length. Suppose the above triagle was right angle (which it isn't), then one could represent Pythagoras' Theorem as a2+b2=c2. This allows you to find any side length of a right-angle triangle, given the other two sides.

Use of these rules in three dimensions usually means calculating the volume of 3-dimensional shapes, finding required lengths using the Sine Rule or Pythagoras' Theorem.

Helpful Links
https://en.wikipedia.org/wiki/Law_of_sines (Wikipedia page with a detailed explanation of the sine rule)

https://www.mathsisfun.com/pythagoras.html

http://www.bevs.k12.oh.us/Downloads/WS%20-%20Pythagorean%20Theorem-blizzard%20bag1.pdf (worksheet with questions on Pythagoras' Theorem)

https://www.examsolutions.net/tutorials/exam-questions-sine-rule/ (Exam questions on the sine rule)