There are three basic trigonometric functions defined for a right-angled triangle:

- sine
- cosine
- tan

Let a, and b be the two sides and c be the hypotenuse. Then the functions takes the following form:

- sin x = opposite / hypotenuse = a /c
- cos x = = adjacent / hypotenuse = b / c
- tan x = opposite / adjacent = a / b

**E.g.1**

In the above triangle, AB = 3cm and x = 30^{0}. Find AC and BC.

sin 30 = 3 /AC

0.5 = 3 / AC

AC = 6cm.

cos 30 = b /AC

0.8660 = b / 6

AC = 5.1cm.

**E.g.2**

In the above triangle, BC = 4cm and x = 60^{0}. Find AC and AB.

cos 60 = 4 /AC

0.5 = 4 / AC

AC = 8cm.

tan 60 = AB /BC

1.7 = AB / 4

AB = 6.8cm.

**E.g.3**

In the above triangle, AB = 3cm and BC = 5cm. Find x.

tan x = 3 / 5 = 0.6

x = tan^{-1}0.6 = 28.6

x = 28.6^{0}

**sin A / a = Sin B /b = Sin C /c **

*Note the blinking letters to see the relationship.*

**E.g.1**

A = 30^{0} ; B = 50^{0}; b = 9cm; find c.

c / sin 50 = 9 / sin 30

c = 9 x sin 50 / sin 30

c = 13.7 cm.

**E.g.2**

A = 110^{0} ; b = 5cm; a = 9cm; find B.

5 / sin B = 9 / sin 110

sin B = 5*sin 110 / 9

B = 31.5^{0}

**c ^{2} = a^{2} + b^{2} - 2abcosC**

**E.g.1**

c

c

c

c= 4.6 cm.

**E.g.2**

c

c

c

c= 13cm.

**E.g.3**

c

c

c

c= 13 cm.

**E.g.4**

8

64 = 49 + 36 - 84cosC

-84cosC = -21

cosC = 0.25

C = 75.5

*Proof:*

A = 1/2 X h X b

sin x = h / a => h = a sinx

A = 1/2 absinx

**E.g.**

A = 1/2 * 4 * 8 * sin 30

A = 8cm

Trigonometric values of major angles, such as 0^{0}, 90^{0}, 180^{0}, 270^{0} and 360^{0} can be remembered
with the aid of the trigonometric curves, if you can visualize them. The three major curves are as follows:

The following image shows how to obtain the values of 0^{0}, 180^{0}, and 360^{0} from the basic definitions

The following image shows how to obtain the values of 90^{0} and 270^{0} from the basic definitions

In addition, the values of 30^{0}, 45^{0} and 60^{0} can be derived in the following way:

The trigonometric values of 45^{0} is calculated from an *right-angled isosceles triangle with each equal side being 1 unit.*

The values of 30^{0} and 60^{0} are calculated from an *equilateral triangle* of each side 2 unit.

The angles are measured from the border between the *first* and the *fourth* quadrants - and anticlockwise, by convention. If it is measured in clockwise, it is considered as negative.

The following animation explicitly demonstrates it:

**Now, in order to complement what you have just learnt, work out the following questions:**

- ABC is a triangle. AB = 6cm and angle ABC = 45
^{0}. Find the perimeter and the area of the triangle. - Prove that Pythagoras Theorem is correct using the Cosine Rule.
- The height of a man is 1.2m. When he looks at the top of a tree, standing 20m away from it, the angle of elevation is 32
^{0}. Calculate the height of the tree. - Points P and Q are due south and West of a pillar of height 20m. The angle of elevation of the top from these points are 22
^{0}and 42^{0}respectively. Find the distance between P and Q. - A ship is sailing a distance of 200 miles from port X to port Y, on a bearing 020
^{0}. At Y, it changes its course on a bearing 110^{0}and reaches port Z, after travelling 360 miles. Find the distance between port X and Port Z. Find the bearing of Z from X as well. - A jet is flying a distance of 120 miles from town P to town Q, on a bearing 030
^{0}. At Q, it changes its course on a bearing 60^{0}and reaches town R, after travelling 240 miles. Find the distance between town P and town R. Find the bearing of R from P as well. - A man is looking down from a cliff at another man on the beach. The angle of
depression is 20
^{0}. The second man walks 200m away from the cliff and the new angle of depression for the man at the top becomes 15^{0}. Find the vertical height of the cliff. - The two sides of a right-angled triangle are 5cm, 12cm. Find its size of hypotenuse and the rest of the angles of the triangle.
- The three sides of a triangle are 8cm, 7cm and 13cm. Find its area. Calculate the shortest distance between a vertex and the longest side as well.
- ABC is a triangle with AB = 4cm and AC = 5cm. AD is at an right angle to BC. AD = 3cm. Find BC.

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level.
They are organized according to these specific levels.

The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.

This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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