Mensuration and trigonometry Study Guide

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible (it will never happen) and 1 means the event is certain (it will always happen). A probability of 0.5 (or 1/2 or 50%) means the event is equally likely to happen or not happen. We often calculate theoretical probabilities for situations like rolling a fair dice or flipping a fair coin, where all outcomes are equally likely. The formula for theoretical probability is: P(event) = (Number of favourable outcomes) / (Total number of possible outcomes).

Relative frequency, also known as experimental probability, is an estimate of probability based on observations from an experiment or a series of trials. It is calculated by dividing the number of times an event occurred by the total number of trials conducted. The formula for relative frequency is: Relative Frequency = (Number of times an event occurs) / (Total number of trials). As the number of trials in an experiment increases, the relative frequency of an event tends to get closer to its true theoretical probability. For example, if you flip a coin 10 times, you might get 7 heads. The relative frequency of heads would be 7/10 = 0.7. If you flip it 1000 times, you would expect the relative frequency to be much closer to 0.5. Relative frequency is particularly useful when theoretical probability cannot be easily determined, such as predicting weather patterns or the reliability of a component.

The Cosine Rule is a fundamental trigonometric formula used to find a missing side length or angle in any triangle when the Sine Rule cannot be applied. It is given by:

• To find a side: a² = b² + c² - 2bc cos(A)
• To find an angle: cos(A) = (b² + c² - a²) / (2bc)

Here, 'a', 'b', 'c' are the lengths of the sides of the triangle, and 'A' is the angle opposite side 'a'. Similarly, 'B' is opposite 'b', and 'C' opposite 'c'. You must know which angle corresponds to which side. For example, if you are looking for side 'a', you must use angle 'A'. The higher methods in this context involve applying the Cosine Rule to more complex geometric problems, possibly involving multiple steps, interpreting diagrams carefully, or using it in conjunction with other geometric theorems.

Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 (impossible event) and 1 (certain event). The basic formula is P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes). Relative frequency is an experimental estimate of probability. It is calculated as (Number of times an event occurs) / (Total number of trials). As the number of trials increases, the relative frequency generally becomes a more accurate estimate of the true probability. Higher methods in probability involve understanding when to apply theoretical probability versus relative frequency, reasoning about outcomes of multiple events, and using probability in multi-stage problems often represented with tree diagrams, or dealing with conditional probabilities indirectly within problem-solving contexts. The key is to justify choices of method and interpret results within given contexts.