Here’s an Introduction to 10th Maths Trigonometry,
As the name suggests, trigonometry is all about triangles.
To be more specific, it is more about right- angled triangles, the triangles with one angle equal to 90 degrees.
It is a system that helps us to find missing angles and missing sides of a triangle.
The word trigono means triangle and the word metry means measure.
The little box in the corner of the triangle denotes the right angle which is equal to 90°.
The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H).
The side that is opposite to the angle θ is known as the opposite(O).
And the side which lies next to the angle is known as the Adjacent(A)
According to Pythagoras theorem, we know that,
In a right-angle triangle,
(Opposite)2+(Adjacent)2= (Hypotenuse)2
The sine, cosine and the tangent are the three basic functions in introduction to trigonometry which shows the relation between all the sides of the triangle.
NAME | ABBREVIATION | RELATIONSHIP |
Sine | Sin | Sin (θ)= Opposite/ Hypotenuse |
Cosine | Cos | Cos (θ)= Adjacent/Hypotenuse |
Tangent | Tan | Tan (θ)= Opposite/Adjacent |
Cosecant | Cosec | Cosec (θ)= Hypotenuse/Opposite |
Secant | Sec | Sec (θ)= Hypotenuse/Adjacent |
Cotangent | Cot | Cot (θ)= Adjacent/Opposite |
Note: It should be noted that Sec (θ) does not refer to the product of Sec and θ. Sec (θ) is correctly read as secant of angle (θ).
Tan (θ)= | Sin(θ) / Cos(θ) |
Sec (θ)= | 1 / Cos(θ) |
Cot ( θ)= | Cos(θ) / Sin(θ) |
Cosec ( θ)= | 1 / Sin(θ) |
Trigonometry identities are the trigonometry equations that comprises the trigonometry ratios of all the angles.
They can be formulated through a right angle triangle.
We can express each trigonometric ratio in the terms of another trigonometric ratio.
If one of the values of trigonometry ratio is known we can find out the other value of the trigonometry ratio easily.
They can also be used to acquire the various trigonometry formulas.
Supposedly we take the measure of the angle in radians which is equal to x.
sin (x) = 1 / Cosec(x) | cosec (x) = 1 / Sin(x) |
co(x) = 1 / Sec(x) | sec(x) = 1 / Cos(x) |
tan (x) = 1 / Cos(x) | cot (x) = 1 / Tan(x) |
1. | (Sin x ) ⨯ (cosec x) =1 |
2. | (cos x) ⨯ (sec x) =1 |
3. | (tan x) ⨯ (cot x) =1 |
The three basic identities included in 10th trigonometry are-
Sin^{2}(ፀ) + Cos^{2}(ፀ) = | 1 |
Tan^{2}(ፀ) + 1 = | Sec^{2}(ፀ) |
Cot^{2}(ፀ) + 1 = | Cosec^{2}(ፀ) |
1. | Sin(-x) = - sin x |
2. | Cos(-x) = cos x |
3. | Tan(-x) = -tan x |
4. | Cot(-x) = -cot x |
5. | Sec(x) = sec x |
6. | Cosec(-x) = -cosec x |
I QUADRANT | II QUADRANT |
Sin ፀ increases from 0 to 1 Cos ፀ decreases from 1 to 0 Tan ፀ increases from 0 to ∞ Cot ፀ decreases from ∞ to 0 Sec ፀ increases from 1 to ∞ Cosec ፀ decreases from ∞ to 1 | Sin ፀ decreases from 1 to 0 Cos ፀ decreases from 0 to -1 Tan ፀ increases from - ∞ to 0 Cot ፀ decreases from 0 to - ∞ Sec ፀ increases from -- ∞ to -1 Cosec ፀ decreases from 1 to ∞ |
III QUADRANT | IV QUADRANT |
Sin ፀ increases from 0 to -1 Cos ፀ decreases from -1 to 0 Tan ፀ increases from 0 to ∞ Cot ፀ decreases from ∞ to 0 Sec ፀ decreases from -1 to - ∞ Cosec ፀ decreases from -∞ to -1 | Sin ፀ increases from -1 to 0 Cos ፀ increases from 0 to 1 Tan ፀ increases from - ∞ to 0 Cot ፀ decreases from 0 to - ∞ Sec ፀ decreases from ∞ to 1 Cosec ፀ decreases from -1 to ∞ |
Question1) Prove the following identity -
sin8 ፀ - cos8 ፀ = (sin2 ፀ - cos2ፀ)(1 - 2sin2ፀ cos2 )
Solution) sin8 ፀ - cos8 ፀ = (sin4 ፀ )2-(cos4 ፀ)2
= (sin4 ፀ - cos4 ፀ) (sin4 ፀ +cos4 ፀ)
= (sin2 ፀ - cos2 ፀ) (sin2ፀ +cos2ፀ) (sin4ፀ +cos4ፀ)
= (sin2ፀ - cos2ፀ) (sin4ፀ +cos4ፀ +2sin2ፀ cos2ፀ -2sin2ፀcos2ፀ)
= (sin2ፀ - cos2ፀ)[ (sin2ፀ+cos2ፀ)2- 2sin2ፀcos2ፀ]
= (sin2ፀ - cos2ፀ)(1 - 2sin2ፀcos2ፀ) Proved.
Question 2) Prove the following identity:
cot4ፀ +cot2ፀ = cosec4ፀ + cosec2ፀ
Solution) (cot2ፀ)2 +cot2ፀ
= (cosec2ፀ - 1)2 +(cosec2ፀ - 1)
Since (cot2ፀ + 1 = cosec2ፀ )
= cosec4ፀ - 2cosec2ፀ + 1 + cosec2ፀ -1
= cosec4ፀ + cosec2ፀ (Proved).
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