Important Formula to Solve Trigonometric Problems

There are three ratios in Trigonometry, total questions of trigonometry is based on these 6 trigonometric ratios, even we can say only three main trigonometric ratios are in the Trigonometry, rest three are the reciprocals of the main trigonometric ratios.

Full Name and Short Name of Trigonometric Ratios:

Sr. No.
Full Name
Short Name
1.
Sine
Sin
2.
Cosine
Cos
3.
Tangent
Tan
4.
Cotangent
Cot
5.
Secant
Sec
6.
Cosecant
Cosec

Basic Formulas of Trigonometry:



















Formula Fort 01:
Hypotenuse^2 = Perpendicular^2 + Base^2
Perpendicular^2 =Hypotenuse^2 -  Base^2
Base^2 = Hypotenuse^2 -  Perpendicular^2

Formula Fort 02:
Sinx = Perpendicular / Hypotenuse
Cosx = Base / Hypotenuse
Tanx = Perpendicular / Base
Cosecx = Hypotenuse / Perpendicular
Secx = Hypotenuse / Base
Cotx = Base / Hypotenuse

Formula Fort 03:
Sinx=1/Cosecx
Sinx.Cosecx=1
Cosx=1/Secx
Cosx.Secx=1
Tanx=1/Cotx
Tanx.Cotx=1
Sin2x+Cos2x=1
Sin2x=1-Cos2x
Cos2x=1-Sin2x
Sec2x-Tan2x=1
Sec2x =1+Tan2x
Sec2x-1=Tan2x
Cosec2x-Cot2x=1
Cosec2x =1+Cot2x
Cot2x= Cosec2x-1
Tanx=Sinx/Cosx
Cotx=Cosx/Sinx

Questions and Shortcut Solutions:

Q1. If Cosx=3/5 then Sinx=?
Shortcut Solution:
Cosx=3/5=Base/Hypotenuse
So, Base=3 and Hypotenuse=5
Since, Perpendicular^2 =Hypotenuse^2 -  Base^2
Find Perpendicular by applying above formula which gives Perpendicular=4
Now, Sinx=Perpendicular / Hypotenuse = 4/5

Q2. If Secø+Tanø=2 then Secø=?
Shortcut Solution:
Sec2ø-Tan2ø=1
(Secø-Tanø) (Secø+Tanø)=1
(Secø-Tanø) 2=1
(Secø-Tanø) =1/2 __________(1)
Also given,
 Secø+Tanø=2  __________(2)
Solve both the equations (1) and (2), we get

Secø = 5/4

Q3. Sec4ø-Sec2ø = ?
Shortcut Solution:
Sec4ø-Sec2ø
=Sec2ø (Sec2ø-1)
=Sec2ø Tan2ø
=(1+ Tan2ø)Tan2ø
=Tan2ø + Tan4ø

Q4. Find the value of Sin2750+Cos2750.
Shortcut Solution:
Since, Sin2ø+Cos2ø=1
so, Sin2750+Cos2750also will be equal to 1.

Q5. What will be the value of Sec21080-Tan21080.
Shortcut Solution:
Since, Sec2ø-Tan2ø=1

so, Sec21080-Tan21080also will be equal to 1.

Q6. Cosec2780-Cot2780=?
Shortcut Solution:
Since, Cosec2ø-Cot2ø=1
so, Cosec2780-Cot2780also will be equal to 1.

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