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Calculating bearings given other angles

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Bearings, mathematics GCSE revision guide, looking at bearings and angles including explanations, examples and videos.

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Basic

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Method 1
Draw a line between the ship and the shark. Draw a line pointing North from the ship.

The shark is more than 180° clockwise from North. Place the centre of the protractor on the ship and 180° facing North.

Measure the angle clockwise from 0° to the line between the ship and the shark. In this case the measurement is 100°.

A bearing is measured from North. Add the angle to 180°:

The bearing of the shark from the ship is 280°.
Method 2
Draw a line between the ship and the shark. Draw a line pointing north from the ship.

The shark is more than 180° from North. Place the centre of a protractor on the ship and 0° facing North.

Measure the angle anticlockwise from 0° to the line between the ship and the shark. In this case the measurement is 80°.

A bearing is measured clockwise from North. Subtract this angle from 360°.

The bearing of the shark from the ship is 280°.

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Bearings - Loci and constructions - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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Method 1 Draw a line between the ship and the shark. Draw a line pointing North from the ship. The shark is more than 180° clockwise from North. Place the centre of the protractor on the ship and 180° facing North. Measure the angle clockwise from 0° to the line between the ship and the shark.

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Standard form including calculations

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Standard form is a way of writing down very large or very small numbers easily. 10 ³ = 1000, so 4 × 10 ³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative. The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

This video explains standard form:
Example
Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10 ¹³

It's 10 ¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19
Example
Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10⁻⁶

It's 10⁻⁶ because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.
Manipulation in Standard Form
This is best explained with an example:
Example
The number p written in standard form is 8 × 10 ⁵

The number q written in standard form is 5 × 10⁻²

Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.
Example
Calculate p ÷ q.

Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits. (8 ÷ 5) × (10 ⁵ ÷ 10⁻²) = 1.6 × 10 ⁷

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Standard Form - Mathematics GCSE Revision

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Standard form is a way of writing down very large or very small numbers easily. 10 3 = 1000, so 4 × 10 3 = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

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The song...!

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Video showing how to set up and use a possibility space

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This video shows how to find the height of an isosceles triangle using isosceles

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Pythagoras Theorem   height of an isosceles triangle

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Look at the first 3 examples

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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!
It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

You can also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !

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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!

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Pages 2 and 4 are useful here. They remind us how to solve inequalities and also how to display your answer on a number line.

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BBC - GCSE Bitesize: Solving inequalities

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A secondary school revision resource for GCSE Maths about foundation level symbols, solving inequalities and graphs

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Linear Equations 1,2,3,6 show you the sorts of solving algebra questions you could get

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SOLVING EQUATIONS

This sections illustrates the process of solving equations of various forms. It also shows you how to check your answer three different ways: algebraically, graphically, and using the concept of equivalence.The following table is a partial lists of typical equations.

LINEAR EQUATIONS - Solve for x in the following equations.

1. x - 4 = 10Solution

2. 2x - 4 = 10Solution

3. 5x - 6 = 3x - 8Solution

4. 2(3x - 7) + 4 (3 x + 2) = 6 (5 x + 9 ) + 3Solution

5. Solution

x Solution

Solution

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[Geometry][Differential Equations]

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This sections illustrates the process of solving equations of various forms. It also shows you how to check your answer three different ways: algebraically, graphically, and using the concept of equivalence.The following table is a partial lists of typical equations. LINEAR EQUATIONS - Solve for x in the following equations.

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This video shows you clearly how to expand brackets, including those with negative numbers

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Use this MyMaths lesson to remind yourself how to complete questions with lots of operations

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Use this page to revise how to simplify indices

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A secondary school revision resource for GCSE Maths about foundation level factors, powers and roots

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Use this My Maths lesson to check your understanding of rounding to significant figures

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Use this My Maths lesson to check your understanding of Rounding to Decimal places