Trigonometry for Class 10

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.

    Now you Might Think What a Right-Angle Triangle is, (image will be updated soon)

  • 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


An introduction to Sine, Cosine and Tangent 

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.


Here are the Three Functions you need to know About (10th Maths Trigonometry)

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).


The Following Relations are Obvious From the Definition of Trigonometric Ratios and are Important for 10th Maths Trigonometry:

                Tan (θ)=

Sin(θ) / Cos(θ)

                Sec (θ)=

1 / Cos(θ)

                Cot ( θ)=

Cos(θ) / Sin(θ)

                Cosec ( θ)=

1 / Sin(θ)


Graphs of the Different Functions of 10th Trigonometry – 


         Graph for sin (θ) (image will be updated soon)

    

  Graph for Cos (θ) (image will be updated soon)

 

 Graph for Tan (θ) (image will be updated soon)


Trigonometry class 10 notes on Trigonometry Identities -

  • 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.


Trigonometry class 10 Notes on Relations Between Trigonometric Functions of Real Numbers-


Reciprocal Relations -

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)


From These Results, it Follows That,

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-

Trigonometric Identities-

Sin2(ፀ) + Cos2(ፀ) =

                  1

Tan2(ፀ) + 1 =

                Sec2(ፀ)

Cot2(ፀ) + 1 =

              Cosec2(ፀ)


Opposite Real Number Identities for Trigonometry class 10 -

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


Variations in Values Of Trigonometric Functions in Different Quadrants (image of quadrants will be uploaded soon)

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 ∞


Questions to be Solved on 10th Maths Trigonometry

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 + cosec2

Solution) (cot2ፀ)2 +cot2

= (cosec2ፀ - 1)2 +(cosec2ፀ - 1)

Since (cot2ፀ + 1 = cosec2ፀ )

= cosec4ፀ  - 2cosec2ፀ   + 1 + cosec2ፀ  -1 

= cosec4ፀ + cosec2ፀ   (Proved).