Trigonometry

Further trigonometry - Higher - Test

To complete this test, you'll need to use some of the following formulas.

The sine rule:

or 

The cosine rule:
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B
c² = a² + b² - 2ab cos C

Area of triangle ABC:
½ ab sin C
or, ½ ac sin B
or, ½ bc sin A

Where necessary, give answers correct to 3 s.f.

In questions 1-3, state whether you would use the sine rule or thecosine rule to answer the question. Don't find the required side or angle.

http://www.bbc.co.uk/bitesize/quiz/q66598195

Histograms

The following table shows the ages of 25 children on a school bus:

 

Age	Frequency
5-10	6
11-15	15
16-17	4
> 17	0

If we are going to draw a histogram to represent the data, we first need to find the class boundaries. In this case they are 5, 11, 16 and 18. The class widths are therefore 6, 5 and 2.

The area of a histogram represents the frequency.

The areas of our bars should therefore be 6, 15 and 4.

Remember that in a bar chart the height of the bar represents the frequency. It is therefore correct to label the vertical axis 'frequency'.

However, as in a histogram, it is the area which represents the frequency.

It would therefore be incorrect to label the vertical axis 'frequency' and the label should be 'frequency density'.

So we know that Area = frequency = Frequency density x class width hence:

Frequency density = frequency ÷ class width

Your SUPER FAVOURITE Surds!!!!!!!!!!!!!!!!!!

https://www.youtube.com/watch?v=07bqa_rEtuU

Basic rules

A surd is a square root which cannot be reduced to a whole number. For example, √4 = 2 is not a surd, as the answer is a whole number.

But √5 is not a whole number. You could use a calculator to find that √5 = 2.236067977... but instead of this we often leave our answers in the square root form, as a surd.

You need to be able to simplify expressions involving surds.

Here are some general rules that you will need to learn.

Now have a look at some questions.

Cumulative frequency

The cumulative frequency is obtained by adding up the frequencies as you go along, to give a 'running total'.

Drawing a cumulative frequency diagram

The table shows the lengths (in cm) of 32 cucumbers.

Before drawing the cumulative frequency diagram, we need to work out the cumulative frequencies. This is done by adding the frequencies in turn.

 

Length	Frequency	Cumulative Frequency
21-24	3	3
25-28	7	10 (= 3 + 7)
29-32	12	22 (= 3 + 7 + 12)
33-36	6	28 (= 3 + 7 + 12 + 6)
37-40	4	32 (= 3 + 7 + 12 + 6 + 4)

The points are plotted at the upper class boundary. In this example, the upper class boundaries are 24.5, 28.5, 32.5, 36.5 and 40.5. Cumulative frequency is plotted on the vertical axis.

There are no values below 20.5cm.

Cumulative frequency graphs are always plotted using the highest value in each group of data, (because the table gives you the total that are less than the upper boundary) and the cumulative frequency is always plotted up a graph, as frequency is plotted upwards.

Cumulative frequency diagrams usually have this characteristic S-shape, called an ogive.

Finding the median and quartiles

When looking at a cumulative frequency curve, you will need to know how to find its median, lower and upper quartiles, and the interquartile range.

By drawing horizontal lines to represent ¹/₄ of the total frequency,¹/₂ of the total frequency and ³/₄ of the total frequency, we can read estimates of the lower quartile, median and upper quartile from the horizontal axis.

Quartiles are associated with quarters. The interquartile range is the difference between the lower and upper quartile.

From these values, we can also estimate the interquartile range: 33 - 28 = 5.

Remember to use the total frequency, not the maximum value, on the vertical axis. The values are always read from the horizontal axis.

Box and whisker plots

A box and whisker plot is used to display information about the range, the median and the quartiles. It is usually drawn alongside a number line, as shown -

Example

The oldest person in Mathsminster is 90. The youngest person is 15.

The median age of the residents is 44, the lower quartile is 25, and the upper quartile is 67.

Represent this information with a box-and-whisker plot.

Solution

The Sine Rule

The Law of Sines (sine rule) is an important rule relating the sides and angles of any triangle (it doesn't have to be right-angled!):

If a, b and c are the lengths of the sides opposite the angles A, B and C in a triangle, then:

 a   =   b  =  c 

sinA    sinB  sinC

If you wanted to find an angle, you can write this as:

sinA =  sinB  = sinC

   a          b          c 

https://www.youtube.com/watch?time_continue=5&v=7szHrc3avac

Try this game. It's genius! : -)

http://www.counton.org/games/mathonaire/