Trigonometry Formulas for Class 10

Trigonometry All Formulas for Maths Class 10

If you are searching for all trigonometry formulas of Maths Class 10 at one place, then you are surely come to the right place. Maths-formula.com brings you all the important Maths formulas for class 10 to help you in your preparation for trigonometry class 10 board examination. These formulas are extremely important from the examinations’ point of view.

Maths is a subject where reasoning and logic are very important. Students should have a clear understanding of the underlying theories, concepts and formulas. They must understand that what are the different formulas and what they mean actually. Only then, they will be able to crack the mathematics questions asked in the board examination. Here, we are providing all the formulas related to trigonometry for maths class 10.

Trigonometry Formulas for Class 10

       1.     There are 6 trigonometric ratios, i.e., sin, cos, tan, cot, sec and cosec.

 

       2.     In right-angled triangle ABC, right-angled at B,

 

         AC² = AB² + BC²                    (Using Pythagoras Theorem)

        AB² = AC² – BC²                  

        BC² = AC² – AB²                  

                    

 Trigonometric ratios in terms of sides of a triangle:

Trigonometric Ratio	Mathematical Value
Sin θ	Perpendicular/Hypotenuse or AB/AC
Cos θ	Base/Hypotenuse or BC/AC
Tan θ	Perpendicular/Base or AB/BC
Cot θ	Base/Perpendicular or BC/AB
Sec θ	Hypotenuse/Base or AC/BC
Cosec θ	Hypotenuse/Perpendicular or AC/AB

 

Tip to remember the above formulas is PBP/HHB.

 

      3.     Six trigonometric ratios are related to each other as follows:

      Sin θ = 1/Cosec θ

      Cos θ = 1/Sec θ

     Tan θ = Sin θ/Cos θ

     Cot θ = Cos θ/Sin θ

     Sec θ = 1/Cos θ

     Cosec θ = 1/Sin θ

      4.     If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of the angle can be easily determined.

 

      5.     The values of trigonometric ratios for some specific angles such as 0°, 30°, 45°, 60° and 90° can be determined easily. The values are given as follows:

 

	0°	30°	45°	60°	90°
Sin	0	1/2	1/√2	√3/2	1
Cos	1	√3/2	1/√2	1/2	0
Tan	0	1/√3	1	√3	Not defined
Cot	Not defined	√3	1	1/√3	0
Sec	1	2/√3	√2	2	Not defined
Cosec	Not defined	2	√2	2/√3	1

 

      6.     The values of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than or equal to 1.

      7.     Trigonometric Sign Functions

sin (−θ) = −sin θ

cos (−θ) = cos θ

tan (−θ) = −tan θ

cosec (−θ) = −cosec θ

sec (−θ) = sec θ

cot (−θ) = −cot θ

 

      8.     Trigonometric ratios of complementary angles are as follows:

 

In case of (90° – θ), (90° + θ), (270° – θ) and (270° + θ), we write the value of sin as cos and the value of cos as sin. We write the value of tan as cot and cot as tan. We write the value of sec as cosec and cosec as sec.

 

In case of (180° – θ), (180° + θ), (360° – θ) and (360° + θ), we write the value of sin as sin, cos as cos, tan as tan, cot as cot, sec as sec and cosec as cosec.

To identify the sign of the value, we have to learn ASTC.

 

Let us understand the meaning of ASTC.

 

A means All positive in Ist quadrant.

S means Sin and its reciprocal Cosec are positive in IInd quadrant and remaining are negative.

T means Tan and its reciprocal Cot are positive in IIIrd quadrant and remaining are negative.

C means Cos and its reciprocal Sec are positive in IVth quadrant and remaining are negative.

 

First Quadrant:

sin (90° – θ) = cos θ;             cos (90° – θ) = sin θ

tan (90° – θ) = cot θ;             cot (90° – θ) = tan θ

sec (90° – θ) = cosec θ;         cosec (90° – θ) = sec θ

 

Second Quadrant:

sin (90° + θ) = cos θ;                cos (90° + θ) = –sin θ

tan (90° + θ) = –cot θ;             cot (90° + θ) = –tan θ

sec (90° + θ) = –cosec θ;         cosec (90° + θ) = sec θ

 

sin (180° – θ) = sin θ;                cos (180° – θ) = –cos θ

tan (180° – θ) = –tan θ;            cot (180° – θ) = –cot θ

sec (180° – θ) = –sec θ;            cosec (180° – θ) = cosec θ

 

Third Quadrant:

sin (180° + θ) = –sin θ;             cos (180° + θ) = –cos θ

tan (180° + θ) = tan θ;             cot (180° + θ) = cot θ

sec (180° + θ) = –sec θ;           cosec (180° + θ) = –cosec θ

 

sin (270° – θ) = –cos θ;            cos (270° – θ) = –sin θ

tan (270° – θ) = cot θ;              cot (270° – θ) = tan θ

sec (270° – θ) = –cosec θ;        cosec (270° – θ) = –sec θ

 

Fourth Quadrant:

sin (270° + θ) = –cos θ;             cos (270° + θ) = sin θ

tan (270° + θ) = –cot θ;             cot (270° + θ) = –tan θ

sec (270° + θ) = cosec θ;           cosec (270° + θ) = –sec θ

 

sin (360° – θ) = –sin θ;              cos (360° – θ) = cos θ

tan (360° – θ) = –tan θ;            cot (360° – θ) = –cot θ

sec (360° – θ) = sec θ;              cosec (360° – θ) = –cosec θ

 

      9.     These are the trigonometric identities:

For  0° ≤ θ ≤ 90°

sin² θ + cos² θ = 1 ;             sin² θ = 1 – cos² θ ;           cos² θ = 1 – sin² θ

sec² θ – tan² θ = 1 ;             sec² θ = 1 + tan² θ ;           tan² θ = sec² θ – 1

cosec² θ – cot² θ = 1 ;          cosec² θ = 1 + cot² θ ;       cot² θ = cosec² θ – 1

 

      10. Double Angle Formulas:

sin 2θ = 2 sin θ cos θ = [2 tan θ /(1 + tan² θ)]

cos 2θ = cos² θ – sin² θ = 1 – 2 sin² θ = 2 cos² θ – 1 = (1 – tan² θ)/(1 + tan² θ)

tan 2θ = (2 tan θ)/(1 – tan² θ)

 

      11. Triple Angle Formulas:

sin 3θ = 3 sin θ – 4 sin³ θ

cos 3θ = 4 cos³ θ – 3 cos θ

tan 3θ = [3 tan θ – tan³ θ]/[1 − 3 tan² θ]

 

      12. Half Angle Formulas:

       sin θ = 2 sin θ/2 cos θ/2 = [2 tan θ/2 /(1 + tan² θ/2)]

       cos θ = cos² θ/2 – sin² θ/2 = 1 – 2 sin² θ/2 = 2 cos² θ/2 – 1 = (1 – tan² θ/2)/(1 + tan² θ/2)

       tan θ/2 = (2 tan θ/2)/(1 – tan² θ/2)