Trigonometric Formulas

There are some special formulas in Trigonometry such as “Transformation Formulas, Inverse Transformation Formulas, Trigonometric Sum and Difference Formulas, Half Angle Formulas”. In this article, we will tell about these special formulas, friends.

 

Sum and Difference Formulas in Trigonometry;

♦  sin (x + y) = sin x. cos y + cos x. sin y

♦  cos (x + y) = cos x. cos y – sin x. sin y

♦  tan (x + y) = ​\( \displaystyle\frac{tan~x + tan~y}{1- tan~x. tan~y} \)

♦  cot (x + y) = ​​\( \displaystyle\frac{1}{tan(x+y)}​ \)

♦  sin (x – y) = sin x. cos y – cos x. sin y

♦  cos (x – y) = cos x. cos y + sin x. sin y

♦  tan (x – y) = ​​\( \displaystyle\frac{tan~x – tan~y}{1+ tan~x. tan~y} \)

♦  cot (x – y) = ​\( \displaystyle\frac{1}{tan(x-y)} \)

 

NOTE : If a, b ∈ ​\( \mathbb{R} \)​;

\( -\sqrt{a^2 + b^2}​≤a.sinx +b.cosx≤\sqrt{a^2 + b^2}​ \)

 

Half Angle Formulas;

♦  sin 2x = 2. sin x. cos x

♦  cos 2x = cos²x – sin²x

♦  cos 2x = 2. cos²x – 1

♦  cos 2x = 1 – 2. sin²x

♦  tan 2x  = ​\( \displaystyle\frac{2. tan x}{1 – tan^2 x} \)

 

Pythagoras Formulas;

♦  sin²x + cos²x = 1

♦  tan²x + 1 =  sec²x

♦  cot²x + 1 =  cosec²x

 

Transformation Formulas in Trigonometry;

♦  ​​\( sin x + sin y = 2.sin\displaystyle\frac{x + y}{2}. cos\displaystyle\frac{x – y}{2}​ \)

♦  ​\( sin x – sin y = 2.cos\displaystyle\frac{x + y}{2}. sin\displaystyle\frac{x – y}{2} ​ \)

♦ ​\( cos x + cos y = 2.cos\displaystyle\frac{x + y}{2}. cos\displaystyle\frac{x – y}{2} \)

♦  ​​\( cos x – cos y = – 2.sin\displaystyle\frac{x + y}{2}. sin\displaystyle\frac{x – y}{2} \)

♦  ​​\( \displaystyle\frac{sin x + sin \displaystyle\frac{x + y}{2} + sin y}{cos x + cos \displaystyle\frac{x + y}{2} + cos y} = tan\displaystyle\frac{x + y}{2} \)

♦  ​​\( \displaystyle\frac{cos x + cos \displaystyle\frac{x + y}{2} + cos y}{sin x + sin \displaystyle\frac{x + y}{2} + sin y} = cot\displaystyle\frac{x + y}{2} \)​​

 

Inverse Transformation Formulas in Trigonometry;

♦  ​​\( cos x. cos y = \displaystyle\frac{1}{2}[cos (x + y) + cos (x – y)] \)

♦  ​​\( sin x. cos y = \displaystyle\frac{1}{2}[sin (x + y) + sin (x – y)] \)

♦  ​​\( sin x. sin y = – \displaystyle\frac{1}{2}[cos (x + y) – cos (x – y)] \)

 

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