Class 8 – Trigonometry
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Introduction: We have seen triangles and their properties, in particular about right triangles in previous classes. We observed different daily life situations where we were using right triangles. Let us look at some of the daily life situations.
Electric poles are present everywhere. They are usually erected by using a metal wire.
The pole, wire and the ground form a triangle. But, if the length of the wire decreases, what will be the shape of the triangle and what will be the angle of the wire with the ground?
In a playground, children like to slide on slider and slider is on a defined angle from the ground. What will happen to the slider if we change the angle? Will children still be able to play on it?
In all situations given above, the distances or heights or angles can be found by using some mathematical techniques, which come under a branch of mathematics called ‘trigonometry’.
The word ‘trigonometry’ is derived from the Greek words ‘tri’ means three, ‘gonia’ means angles and ‘metron’ means measure. Thus, trigonometry is the study of relationships between the sides and angles of a triangle.
Naming the sides in a right triangle: let us take a right triangle ABC as show in the figure. In triangle ABC, we can consider ∠CAB = ∠A = θ where ∠A is an acute angle. Since AC is the longest side, it is called “hypotenuse”. Here you observe the position of side BC with respect to ∠A. It is opposite to ∠A and we can call it as “opposite side of ∠A”. And the remaining side AB can be called as “Adjacent side of ∠A”
BC =Opposite side of ∠A
AB =Adjacent side of ∠A.
Trigonometric ratio: We have seen the examples in the beginning of the chapter which are related to our daily life situations. Let’s know about the trigonometric ratios and how they are defined.
1. Draw a horizontal line on a paper.
2. Let the initial point be A and mark other points B, C, D and E at a distance of 3 cm, 6 cm, 9 cm, 15 cm respectively from A.
3. Draw the perpendiculars BP, CQ, DR and ES of lengths 4 cm, 8 cm, 12cm, 16 cm, from the points B, C, D and E respectively.
4. Then join AP, PQ, QR and RS.
5. Find length of AP, AQ, AR and AS.
Length of hypotenuse
Length of opposite side
Length of adjacent side
Then find the ratios of .
Did you get the same ratio?
Similarly try to find the ratios What do you observe? We get the ratio same.
Defining trigonometric ratios: In the above activity, when we observe right angle triangles ABP, ACQ, ADR and AES, is common, are right angles and are also equal. Hence, we can say that triangles ABP, ACQ, ADR and AES are similar triangles. When we observe the ratio of opposite side of and hypotenuse in a right angle triangle and the ratio of similar sides in another triangle, it is found to be constant in all above right angle triangles ABP, ACQ, ADR and AES and the ratios can be named as “sine A” or simply “sin A” in those triangles. If the value of is “x” when it was measured, then the ratio would be “sin x”.
Hence, we can conclude that the ratio of opposite side of an angle (measure of the angle) and length of the hypotenuse is constant in all similar right angle triangles. This ratio will be named as “sine of that angle”.
Similarly, when we observe the ratios ,it is also found to be constant. And these are the ratios of the adjacent sides of the and hypotenuses in right angle triangles ABP, ACQ, ADR and AES. So, the ratios will be named as “cosineA” or simply “cos A” in those triangles. If the value of the is
“x”, then the ratio would be “cos x”.
Hence, we can also conclude that the ratio of the adjacent side of the angle (measure of the angle) and length of the hypotenuse is constant in all similar right angled triangles. This ratio will be named as “cosine” of that angle.Similarly, the ratio of opposite side and adjacent side of an angle is constant and it can be named as “tangent” of that angle and will be named as “tanA”.
Let’s define ratios in a right angle triangle: Consider a right angle triangle ABC having right angle at B as shown in the following figure. Then, trigonometric ratios of the in right angle triangle ABC are defined as follows:
There are three more ratios defined in trigonometry which are considered as multiplicative inverse or reciprocals of the above three ratios.
Multiplicative inverse of “sine A” is “cosecant A”. Simply written as “cosec A”(or) “csc A”
Similarly, multiplicative inverse of “cos A” is “secant A” (simply written as “sec A”) and that of “tan A” is “cotangent A “(simply written as cot A) i.e.,
How can you define ‘cosec’ in terms of sides?
Example: Given ,find the other trigonometric ratios of the A.
Solution: Let us first draw a right ∆ ABC
Now, we know that .
Therefore, if BC = 4k, then AB = 3k is a positive number.
Now, by using the pythagoras Theorem, we have
So, now we can write all the trigonometric ratios using their definitions.
Trigonometry is an analytical study of a three angled gemetric figure – namely the triangle.
Hipparchus (140 B.C.), a Greek mathematician established the relationships between the sides and angles of a triangle. Greek trigonometry was futher developed by hindu mathematicians. They replaced the chords used by the greeks by half chords of circles with given radii i.e., the equivalent of our sine functions. The study of trigonometry is of great importance in several fields.
Example: In surveying Astronomy, Navigation and Engineering.