Hệ thức cơ bản lượng giác dùng cho 101112

I Hệ thức cơ bản

 Sin2 x + cos2 x = 1

 1 + tan2 x =

1 cos2x

 1 + cot2x =

1 sin2x

 tanx.cotx = 1

 tan x =

sin x cos x

 cot x =

cos x sin x

Hai góc (cung)

đối nhau: Hai góc (cung) bùnhau: Hai góc (cung) phụnhau: Hai góc (cung) hơnkém π :

Hai góc (cung) hơn kém

π

2 :

 cos (-x)= cos x

 sin (-x) = -sin x

 tan (-x) = -tan x

 cot (-x) = -cot x

 sin ( π - x) = sin x

 cos ( π - x)= -cos x

 tan ( π - x) =-tan x

 cot ( π - x) = -cot x

 sin (

π

2 - x) = cos x

 cos (

π

2 - x) = sin x

 tan (

π

2 - x) = cot x

 cot (

π

2 - x) = tan x

 sin ( π +x) =–sin x

 cos ( π +x) =–cos x

 tan ( π +x) =tan x

 cot ( π +x) =cot x

 sin(

π

2 +x) =cos x

 cos (

π

2 +x) = -sin x

 tan (

π

2 +x) = –cot x

 cot (

π

2 +x) = –tan x

 Sin (a+b) sinacosb + sinbcosa

 Cos (a+b) cosacosb - sinasinb

 Tan (a+b)¿ tana+tanb

1−tanctanb

 Cot (a+b) ¿cotacotb−1

cota+ cotb

 Sin (a-b) = sinacosb - sinbcosa

 Cos (a-b) = cosacosb + sinasinb

 Tan (a-b)¿ tana−tanb

1+tanctanb

 Cot (a-b) ¿cotacotb+1

cota−cotb

Nhân đôi: Hạ bậc: Tính sin2 a, cos2 a, tan2 a, cot2 a theo cos2a:

 Cos 2a =cos2 a – sin2 a = 2cos2 a – 1= 1- 2sin2 a

 Sin 2a = 2sinacosa

 Tan 2a = 2 tan a

1−tan2a

 Sin 2 a = 1+ cos2 a

2

 Cos2 a = 1−cos2 a2

 Tan2 a =1−cos2 a 1+cos2 a

Nhân ba:

Tính cos a, sin a, tan a theo t = tan

a

2

 Cos 3a = 4cos3 a – 3cos a

 Sin 3a = 3sin a – 4sin3 a cos a =1−t

2

1+t2tan a =

2 t 1−t2

sin a = 2t

1+ t2

Tích thành tổng:

 Cos a.cos b = 1

2[cos (a+b) + cos (a-b)]

 Sin a.sin b = 1

2[cos (a-b) - cos (a+b) ]

 Sin a.cos b = 1

2[sin (a+b) + sin (a-b)]

Tổng thành tích:

 sin a+ b=2 sin1

2(a+b ) cos

1

2(a−b)

 cos α+cos b=2 cos1

2(α +b) cos

1

2(α −b )

 tan a +¿tan b = sin(a+b)

cosa cosb

 cot a +¿ cot b = sin(a+b)

sin a sin b

sin a−b=2 cos1

2(a+b )sin

1

2(a−b)

cos α−cosb=−2 sin1

2( α +b )sin

1

2(α −b )

 tan a −¿tan b = sin(a−b)

cosa cosb

cot a −¿ cot b = sin(b−a)

sin a sin b

 sin x

+ cos x = √2sin (x +

π

4 ) = √2cos (x −¿

π

4 )

 sin x−¿

cos x = √2sin (x −¿

π

4 ) = −¿√2 cos (x +

π

4 )

 sin x + √3 cos = 2sin (x +

π

3 ) = 2 cos (x −¿

π

6 )

 sin x −¿ √3 cos = 2sin (x -

π

3 ) = −¿2 cos (x +

π

6 )

 √3

sin x + cos x = 2sin (x +

π

6 ) =2 cos (x −¿

π

3 )

 √3

sin x −¿ cos x = 2sin (x −¿

π

6 ) = −2 cos (x +

π

3 )