MATHEMATICS: FORM TWO: Topic 9 – TRIGONOMETRY

Trigonometry is all about Triangles. In this chapter we are going to deal with Right Angled Triangle. Consider the Right Angled triangle below:

The sides are given names according to their properties relating to the Angle .

Adjacent side is adjacent (next to) to the Angle

Opposite side is opposite the Angle

Hypotenuse side is the longest side

Sine, Cosine and Tangent of an Angle using a Right Angled Triangle

Define sine, cosine and tangent of an angle using a right angled triangle

The special Angles we are going to deal with are 30⁰, 45⁰, 60⁰, 90⁰. Let us see how to get the Tangent, Sine and Cosine of each angle as follows:

First, consider an equilateral triangle ABC below, the altitude from C bisects at D.

AD = BD = 1 (bisection)

The results above can be summarized in table as here below:

Find the value of x if

We can find the trigonometrical ratio of any angle by reading it on a trigonometrial table in the same way as we did in reading logarithm of a number on a logarithimic table.

The angle is read from the extreme left hand column and then the corresponding value under the corresponding column of minutes and seconds whenever there is seconds. If we are given angle with zero minute (0’), we read the corresponding value of an angle under the column labeled 0’.

For example; if we are to find the sin56⁰, we have to go to the column extreme to the left. Run your finger down until you meet56⁰, then slide your finger to the exactly same raw to the column labeled 0’. The answer will is 0.8290.

Another example: find cos 78⁰45′. Read the angle78⁰ to the column extreme to the left and then slide your figure to the exactly same angle until you meet the column labeled 45’. The table I’m using has no 45’, so, I have to read the number near to 45’. This number is 42’. The answer of cos78⁰42’is 0.1959. the minutes remained, we are going to read them to the difference columns. Slide your figure to the same column of degree78⁰to the difference column labeled 3’ (minutes remained). The answer is 9. But the instructions says, ‘numbers to the difference columns to be subtracted, not added’. This means we have to subtract 9 (0.0009) from 0.1959. When we subtract we remain with 0.1950. Therefore, cos78⁰45’= 0.1950.

Note that, you can read in the same way the tangent of an angle as we read cosine and sine of an angle. Make sure you read the tables of Natural sine or cosine and or tangent and not otherwise.

Use table to find the value of:

1. sin 55⁰
2. cos 34.4⁰
3. tan 60.2⁰

Angle of Elevation of an Object as seen by an Observer is the angle between the horizontal and the line from the Object to the Observer’s aye (the line of sight). See the figure below for better understanding

The angle of Elevation of the Object from the Observer is α⁰.

Angle of depression of an Object which is below the level of Observer is the angle between the horizontal and the Observer’s line of sight. To have the angle of depression, an Object must be below the Observer’s level. Consider an illustration below:

The angle of depression of the Object from the Object is β⁰

From the top of a vertical cliff 40 m high, the angle of a depression of an object that is level with the base of the cliff is35⁰. How far is the Object from the base of the cliff?