Mensuration and trigonometry Revision Notes

The Sine Rule Formula

The Sine Rule is a powerful tool for solving non-right-angled triangles. For any triangle ABC, with sides a, b, c opposite angles A, B, C respectively, the rule is given by:

To find a side: a/sin(A) = b/sin(B) = c/sin(C)

To find an angle: sin(A)/a = sin(B)/b = sin(C)/c

You must have at least one complete 'pair' (a side and its opposite angle) to use the Sine Rule. For example, if you know side 'a' and angle 'A', you can use this pair with another known side 'b' to find angle 'B', or with another known angle 'B' to find side 'b'.

Finding an Unknown Side

When you need to find the length of an unknown side, you should use the form a/sin(A) = b/sin(B). Arrange the formula so the unknown side is in the numerator.

For example, if you want to find side 'x' opposite angle 'X', and you know side 'y' and angle 'Y', you would set up the equation as x/sin(X) = y/sin(Y). To solve for x, multiply both sides by sin(X): x = (y * sin(X)) / sin(Y). Ensure your calculator is in degree mode if angles are given in degrees. This application is straightforward as side lengths are always positive.

Finding an Unknown Angle and the Ambiguous Case

When finding an unknown angle, use the form sin(A)/a = sin(B)/b. Arrange the formula so the sine of the unknown angle is in the numerator. For example, to find angle 'X': sin(X)/x = sin(Y)/y. This gives sin(X) = (x * sin(Y)) / y. Then, calculate X = arcsin((x * sin(Y)) / y).

Crucially, when using arcsin (sin⁻¹), remember there might be two possible solutions for the angle within 0° and 180°. If sin(X) = k, then X can be arcsin(k) or 180° - arcsin(k). This is known as the 'ambiguous case'. You must check if both solutions are geometrically possible within the triangle (i.e., if the sum of all angles remains below 180°). The ambiguous case typically arises when you are given two sides and a non-included angle (SSA).

Higher-Level Applications of the Sine Rule

In more complex problems, the Sine Rule might be part of a multi-step solution. This could involve combining it with the Cosine Rule, the formula for the area of a triangle (Area = 0.5ab sin(C)), or even properties of sectors and segments of circles. You might need to first calculate a missing angle using the sum of angles in a triangle (180°) before applying the Sine Rule. Always clearly label diagrams and identify the known pairs and what you need to find. Careful working and attention to detail are crucial, especially when dealing with multiple steps or potential ambiguous cases.

The Probability Scale

Probability measures the chance of an event happening. It is represented by a number from 0 to 1. An event with a probability of 0 is impossible, and an event with a probability of 1 is certain to occur. Probabilities can also be expressed as fractions, decimals, or percentages. For example, a probability of 0.5 means a 50% chance, and a probability of 1/4 means a 25% chance. The sum of probabilities of all possible outcomes for any event must always equal 1. For instance, if the probability of rain is 0.3, the probability of no rain is 1 - 0.3 = 0.7.

Basic Probability Calculation

For situations where all outcomes are equally likely, the probability of an event (P(Event)) is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. For example, when drawing a card from a standard 52-card deck, the probability of drawing a Queen is 4 (number of Queens) / 52 (total cards) = 1/13. This formula is fundamental for theoretical probability, where we know all possible outcomes and their likelihoods perfectly. Always simplify fractions where possible.

Relative Frequency (Experimental Probability)

Relative frequency is an estimate of probability based on observations from an experiment. It is calculated by dividing the number of times an event occurs by the total number of trials conducted. For example, if a biased coin is flipped 200 times and lands on heads 120 times, the relative frequency of heads is 120/200 = 0.6. Critically, as the number of trials in an experiment increases, the relative frequency tends to get closer to the true, theoretical probability of the event. This makes relative frequency a powerful tool for estimating probabilities in real-world situations where the theoretical probability may not be easily determined or known.

Distinction and Application

It's important to differentiate between theoretical probability and relative frequency. Theoretical probability is what we expect based on an ideal model (e.g., a fair coin). Relative frequency is what we observe from actual experiments. Relative frequency is particularly useful when dealing with situations involving chance where the outcomes are not equally likely or the exact probabilities are unknown, such as predicting the probability of a bus being late based on past data. The more data collected through trials, the more reliable the relative frequency becomes as an estimate of the true probability.

The Probability Scale

Probability is a numerical measure of the likelihood of an event. It always lies between 0 and 1, inclusive. An event with a probability of 0 is impossible, meaning it will never happen. An event with a probability of 1 is certain, meaning it will always happen. A probability of 0.5 indicates an event is equally likely to happen or not happen. Probabilities can be expressed as fractions, decimals, or percentages. For example, P = 1/4, P = 0.25, or P = 25% all represent the same likelihood.

Calculating Theoretical Probability

For situations where all outcomes are equally likely, you can calculate the theoretical probability of an event. The formula for this is: P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes). For instance, if you roll a standard six-sided die, the probability of rolling a '4' is 1/6 because there is one '4' and six possible outcomes. The probability of rolling an even number is 3/6 = 1/2 because there are three even numbers (2, 4, 6) out of six possible outcomes.

Understanding Relative Frequency

Relative frequency is an estimate of probability based on observed data from an experiment. It is calculated as: Relative Frequency = (Number of times the event occurs) / (Total number of trials). For example, if a coin is flipped 50 times and lands on heads 28 times, the relative frequency of heads is 28/50 = 0.56. Unlike theoretical probability, relative frequency can change with each set of trials. It is particularly useful for modelling real-world events where all outcomes might not be equally likely or known beforehand.

Relative Frequency and Probability (Law of Large Numbers)

The more trials you perform in an experiment, the closer the relative frequency of an event is likely to get to its true theoretical probability. This concept is often referred to as the Law of Large Numbers. For example, if you flip a fair coin a few times, the relative frequency of heads might be far from 0.5. However, if you flip it thousands of times, the relative frequency of heads will almost certainly be very close to 0.5. Therefore, a higher number of trials generally leads to a more reliable estimate of probability using relative frequency.

The Cosine Rule: Finding Sides

The Cosine Rule is an extension of Pythagoras' theorem and is used in non-right-angled triangles. If you know two sides and the included angle, you can find the third side. The formula is: a² = b² + c² - 2bc cos(A). Remember that 'a' is the side opposite angle 'A', 'b' is opposite 'B', and 'c' is opposite 'C'. For example, if you are given side b, side c, and angle A (the angle between b and c), you can find side a. Ensure your calculator is in degree mode for angle calculations.

The Cosine Rule: Finding Angles

If you know all three sides of a triangle, you can use a rearranged form of the Cosine Rule to find any of the angles. The formula to find angle A is: cos(A) = (b² + c² - a²) / (2bc). To find A, you must then calculate A = arccos((b² + c² - a²) / (2bc)). Make sure you correctly identify which side is 'a', 'b', and 'c' relative to the angle you are trying to find. For instance, 'a' must always be the side opposite the angle 'A' you are looking for.

Relative Frequency as an Estimate of Probability

Relative frequency is an experimental probability calculated by dividing the number of times an event occurs by the total number of trials. For example, if a coin is flipped 100 times and lands on heads 53 times, the relative frequency of heads is 53/100 or 0.53. This is an estimate of the true probability. The more trials conducted, the more reliable and accurate this estimate becomes. This concept allows us to make predictions based on experimental data, especially when theoretical probabilities are difficult to determine.

Applying Combined Probability and Higher Reasoning

Higher methods in probability often involve combining events or dealing with scenarios where the initial probability changes. This might include using tree diagrams for sequential events, understanding conditional probability, or using set notation for intersections and unions of events. It is crucial to reason about the independence of events and whether the outcome of one event affects the probability of another. Always present probabilities as fractions, decimals, or percentages between 0 and 1 (inclusive), unless the question specifies otherwise.