In this post I have list down all the important trigonometric formula which are required to solve problems of grade 11. You need to memorize all the given formula because you need them to solve textbook questions.
The best strategy to remember the formulas is to repeatedly write them in the textbook. I have personally experienced that writing any text help us to remember for long period of time.
Trigonometry Formulas for Class 11
Formulas for (x +y)
1. cos (x + y) = cos x cos y – sin x sin y
2. cos (x – y) = cos x cos y + sin x sin y
3. sin (x + y) = sin x cos y + cos x sin y
4. sin (x – y) = sin x cos y – cos x sin y
5. tan\quad (x+y)\quad =\frac { tan(x)\quad +\quad tan(y) }{ 1\quad -\quad tan(x)\quad tan(y) } \ \
6. tan\quad (x-y)\quad =\frac { tan(x)\quad -\quad tan(y) }{ 1\quad +\quad tan(x)\quad tan(y) }
7. cot\quad (x+y)\quad =\frac { cot(x)\quad cot(y)\quad -\quad 1 }{ cot(y)\quad +\quad cot(x) }
8. cot\quad (x-y)\quad =\frac { cot(x)\quad cot(y)\quad +\quad 1 }{ cot(y)\quad -\quad cot(x) }
Trick to remember (x + y) formulas
a. For cos(x + y) and cos (x – y) formula
we have cos cos and then sin sin in the equation
b. For sin(x + y) and sin (x – y) formula
we have sin cos and then cos sin in the equation
c. For tan (x + y) and tan (x -y) formula
Function tan (x) and tan (y) are in addition form in numerator
and both function are in multiplication form in denominator
d. For cot (x + y) and cot (x -y) formula
Function cot (x) and cot (y) are in multiplication form in numerator
and both function are in addition form in denominator
Formula for 2x
(1) \quad sin(2x)=2sin(x)cos(x) = \frac { 2tan (x) }{ 1+{ tan }^{ 2 }x }
(2) \quad cos(2x)=cos^{ 2 }x -sin^{ 2 }x \quad \\\ \\ \quad \quad \quad \quad =2cos^{ 2 }x -1\quad \\\ \\ \quad \quad \quad \quad =1-2sin^{ 2 }x\quad \\\ \\ \quad \quad \quad \quad =\quad \frac { 1-{ tan }^{ 2 }x }{ 1 + { tan }^{ 2 }x }
There are multiple form of cos(2x) formula, you have to remember each of them.
(3) \quad tan(2x)=\quad \frac { 2\quad tan(x) }{ 1-{ tan }^{ 2 }x }
All the three formulas are extremely important. Please make a point to write multiple times in a note book, this way you will remember the formula for long time.
Formula for 3x
1.\quad sin(3x)\quad =\quad 3sin(x)\quad -\quad 4{ sin }^{ 3 }(x)\\\ \\ \\\ \\ 2.\quad cos(3x)\quad =\quad 4{ cos }^{ 3 }(x)\quad -\quad 3cos(x)\\\ \\ \\\ \\ 3.\quad tan(3x)\quad =\quad \frac { 3tan(x) - { tan }^{ 3 }(x)\quad }{ 1 - 3{ tan }^{ 2 }(x)\quad } \ \Note:
In sin(3x) formula the cube expression comes as second term
while cos(3x) formula is opposite, here the cube expression comes first in the formula.
So you don’t need to remember both the formula, just remember the sin(3x) formula and keep in mind that the cos(3x) expression is just opposite
Addition of Trigonometry Functions
1.\quad cos(x) +cos(y)= 2cos\frac { x+y }{ 2 } cos\frac { x-y }{ 2 } \\\ \\ \\\ \\ 2.\quad cos(x) -cos(y)= -2sin\frac { x+y }{ 2 } sin\frac { x-y }{ 2 } \\\ \\ \\\ \\ 3.\quad sin(x) +sin(y) =2sin\frac { x+y }{ 2 } cos\frac { x-y }{ 2 } \\\ \\ \\\ \\ 4.\quad sin(x) - sin(y) = 2 cos\frac { x+y }{ 2 } sin\frac { x-y }{ 2 } \Note:
a. For cos (x) + cos (y), we have cos, cos in the formula
For cos (x) – cos (y), we have sin, sin in the formula
b. For sin (x) + sin (y), we have sin, cos in the formula
For sin (x) – sin (y), we have cos, sin in the formula
Multiplication of Trigonometry
Following are the formulas involving multiplications of two trigonometric functions
(i) 2 cos x cos y = cos (x + y) + cos (x – y)
(ii) –2 sin x sin y = cos (x + y) – cos (x – y)
(iii) 2 sin x cos y = sin (x + y) + sin (x – y)
(iv) 2 cos x sin y = sin (x + y) – sin (x – y)
Note
a. In multiplication of similar functions [cos(x) * cos(y)] or [sin(x) * sin(y)] ;
there always with two cos functions ==> cos (x+y) +/- cos (x-y)
b. In case of multiplication of different function [sin(x) * cos(y)] or [cos(x) * sin(y)]
there always with two sin functions ==> sin(x+y) +/- sin (x-y)
Trigonometry Solved Questions
In this section we will discuss important trigonometry problems involving above mentioned formulas. These questions will basically give you ideas regarding how to use trigonometry formulas for problem solving.
(01) Find the value of sin 15°
Solution
sin 15° = sin (45 – 30)
Using the formula
sin (x – y) = sin x cos y – cos x sin y
(02) Prove the following
\frac { sin(x+y) }{ sin(x-y) } =\frac { tan(x) + tan(y) }{ tan(x) - tan(y) }
Solution
Taking\quad L.H.S\\\ \\ \frac { sin(x+y) }{ sin(x-y) } =\frac { sin(x).cos(y) + cos(x).sin(y) }{ sin(x)\quad cos(y) - cos(x)\quad sin(y) } \\\ \\ \\\ \\ \ dividing\quad numerator\quad and\quad denominator\\\ \\ \quad by\quad cos(x).cos(y)\\\ \\ \\\ \\ \frac { sin(x+y) }{ sin(x-y) } =\frac { \frac { sin(x).cos(y) }{ cos(x). cos(y) } \quad + \frac { cos(x).sin(y) }{ cos(x). cos(y) } }{ \frac { sin(x)cos(y) }{ cos(x).cos(y) } - \frac { cos(x) sin(y) }{ cos(x). cos(y) } } \\\ \\ \\\ \\ \ \frac { sin(x+y) }{ sin(x-y) } =\frac { tan(x) + tan(y) }{ tan(x)- tan(y) } \ \