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Trigonometry formula

Basic trigonometry formula



  • Perp./Hyp. = sin θ

  • Base/Hyp. = cos θ

  • Perp./Base = tan θ

  • Base/Perp. = cot θ

  • Hyp./Base = sec θ

  • Hyp./Perp. = cosec θ


Baudhayana formula


( Hyp. )2 = ( Base )2 + ( Perp.)2


Some common trigonometric Formulas



  • sin(A+B) = sin A cos B + cos A sin B

  • sin(A-B) = sin A cos B - cos A sin B

  • cos(A+B) = cos A cos B - sin A sin B

  • cos(A-B) = cos A cos B + sin A sin B


  • sin 2A = 2 sin A cos B

  • cos 2A = cos2 A - sin2 A

  • cos 2A = 1 - 2 sin2 A

  • sin(A+B) + sin(A-B) = 2 sin A cos B

  • sin(A+B) - sin(A-B) = 2 cos A sin B

  • cos(A+B) + cos(A-B) = 2 cos A cos B

  • cos(A+B) - cos(A-B) = -2 sin A sin B


Trigonometric Ratios


Trigonometry ratio table:


Table of trigonometrical ratios of some standard angels:














































































Angle

sin θ

cos θ

tan θ

00

0

1

0

300

$$\frac{1}{2}$$

$$\frac{\sqrt{3}}{2}$$

$$\frac{1}{\sqrt{3}}$$

450

$$\frac{1}{\sqrt{2}}$$

$$\frac{1}{\sqrt{2}}$$

1

600

$$\frac{\sqrt{3}}{2}$$

$$\frac{1}{2}$$

$$\sqrt{3}$$

900

1

0

$$\infty $$

1200

$$\frac{\sqrt{3}}{2}$$

$$-\frac{1}{2}$$

$$-\sqrt{3}$$

1350

$$\frac{1}{\sqrt{2}}$$

$$-\frac{1}{\sqrt{2}}$$

-1

1500

$$\frac{1}{2}$$

$$-\frac{\sqrt{3}}{2}$$

$$-\frac{1}{\sqrt{3}}$$

1800

0

-1

0

2700

-1

0

$$-\infty $$

3600

0

1

0














































































Angle

cot θ

sec θ

cosec θ

00

$$\infty $$

1

$$\infty $$

300

$$\sqrt{3}$$

$$\frac{2}{\sqrt{3}}$$

2

450

1

$$\sqrt{2}$$

$$\sqrt{2}$$

600

$$\frac{1}{\sqrt{3}}$$

2

$$\frac{2}{\sqrt{3}}$$

900

0

$$\infty $$

1

1200

$$-\frac{1}{\sqrt{3}}$$

-2

$$\frac{2}{\sqrt{3}}$$

1350

-1

$$-\sqrt{2}$$

$$\sqrt{2}$$

1500

$$-\sqrt{3}$$

$$-\frac{2}{\sqrt{3}}$$

2

1800

$$-\infty $$

-1

$$\infty $$

2700

-1

0

$$\infty $$

3600

$$\infty $$

1

$$\infty $$


Relation between Trigonometric Ratios



  • sin θ cosec θ = 1

  • cos θ sec θ = 1

  • tan θ cot θ = 1

  • tan θ = sin θ/cos θ

  • cot θ = cos θ/sin θ

  • sin2 θ + cos2 θ = 1

  • 1 + tan2 θ = sec2 θ

  • 1 + cot2 θ = cosec2 θ





A. Trigonometric Ratios of acute angles



  • Perp./Hyp. = sin θ

  • Base/Hyp. = cos θ

  • Perp./Base = tan θ

  • Base/Perp. = cot θ

  • Hyp./Base = sec θ

  • Hyp./Perp. = cosec θ




B. Trigonometric ratios of allied angles



1. Trigonometric ratios of (-θ) in terms of (θ)


sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ) = -cosecθ




2. Trigonometric ratios of (900-θ) in terms of (θ)


sin(900-θ) = cosθ
cos(900-θ) = sinθ
tan(900-θ) = cotθ
cot(900-θ) = tanθ
sec(900-θ) = cosecθ
cosec(900-θ) = secθ





3. Trigonometric ratios of (900+θ) in terms of (θ)


sin(900+θ) = cosθ
cos(900+θ) = -sinθ
tan(900+θ) = -cotθ
cot(900+θ) = -tanθ
sec(900+θ) = -cosecθ
cosec(900+θ) = secθ




4. Trigonometric ratios of (1800-θ) in terms of (θ)


sin(1800-θ) = sinθ
cos(1890-θ) = -cosθ
tan(1800-θ) = -tanθ
cot(1800-θ) = -cotθ
sec(1800-θ) = -secθ
cosec(1800-θ) = cosecθ




5. Trigonometric ratios of (1800+θ) in terms of (θ)


sin(1800+θ) = -sinθ
cos(1800+θ) = -cosθ
tan(1800+θ) = tanθ
cot(1800+θ) = cotθ
sec(1800+θ) = -secθ
cosec(1800+θ) = -cosecθ



6. Trigonometric ratios of (900+θ) in terms of (θ)


sin(2700-θ) = -cosθ
cos(2700-θ) = -sinθ
tan(2700-θ) = cotθ
cot(2700-θ) = tanθ
sec(2700-θ) = -cosecθ
cosec(2700θ) = -secθ


7. Trigonometric ratios of (900+θ) in terms of (θ)


sin(2700+θ) = -cosθ
cos(2700+θ) = sinθ
tan(2700+θ) = -cotθ
cot(2700+θ) = -tanθ
sec(2700+θ) = cosecθ
cosec(2700+θ) = -secθ




8. Trigonometric ratios of (3600-θ) in terms of (θ)


sin(3600-θ) = -sinθ
cos(3600-θ) = cosθ
tan(3600-θ) = -tanθ
cot(3600-θ) = -cotθ
sec(3600-θ) = secθ
cosec(3600-θ) = -cosecθ




9. Trigonometric ratios of (3600-θ) in terms of (θ)


sin(3600+θ) = sinθ
cos(3600+θ) = cosθ
tan(3600+θ) = tanθ
cot(3600+θ) = cotθ
sec(3600+θ) = secθ
cosec(3600+θ) = cosecθ




10. Trigonometric ratios of (n×3600±θ) in terms of (θ)


sin(n×3600±θ) = ±sinθ
cos(n×3600±θ) = cosθ
tan(n×3600±θ) = ±tanθ
cot(n×3600±θ) = ±cotθ
sec(n×3600±θ) = secθ
cosec(n×3600±θ) = ±cosecθ



C. Trigonometric ratios of compound angels



1. Trigonometric ratios of sum and difference of two angles


  • sin(A+B) = sinA cosB + cosA sinB

  • cos(A+B) = cosA cosB - sinA sinB

  • sin(A-B) = sinA cosB - cosA sinB

  • cos(A-B) = cosA cosB + sinA sinB




2. Transformation of product into sums of differences


  • 2 sinA cosB = sin(A+B) + sin(A-B)

  • 2 cosA sinB = sin(A+B) - sin(A-B)

  • 2 cosA cosB = cos(A+B) + cos(A-B)

  • 2 sinA sinB = cos(A+B) - cos(A-B)




3. Transformation of sum or difference into product


Suppose A+B=C and A-B=D
or $$ A = \frac{C+D}{2}$$ and $$B = \frac{C-D}{2}$$




  • $$sinC+sinD = 2 sin \frac{C+D}{2} cos\frac{C-D}{2}$$


  • $$sinC-sinD = 2 cos \frac{C+D}{2} sin\frac{C-D}{2}$$


  • $$cosC+cosD = 2 cos \frac{C+D}{2} cos\frac{C-D}{2}$$


  • $$cosC-cosD = 2 sin\frac{C+D}{2} sin\frac{D-C}{2}$$




4. Trigonometric ratios of sum of more than two angles


  • sin(A+B+C) = sinA cosB cos C + cosA sinB cosC + cosA cosB sinC - sinA sinB sinC


  • cos(A+B+C) = cosA cosB cosC - sinA sinB cosC - sinA cosB sinC - cosA sinB sinC




D. Trigonometric ratios of multiple and sub-multiple angles


Multiple angles: 2A, 3A, 4A ......


Sub-multiple angles : $$ \frac{A}{2}, \frac{A}{3}, \frac{A}{4}$$.......


1. Trigonometric ratios of an angle 2A in terms of angle A

  • sin2A = 2sinA cosA


  • cos2A = 1-2sin2A


  • $$ \ tan2A = \frac{2tanA}{1-tan^2A}$$



2. Trigonometric ratios of sin2A and cos2A in terms of tanA


  • $$ \ sin2A = \frac{2tanA}{1+tan^2A}$$


  • $$ \ cos2A = \frac{1-tan^2A}{1+tan^2A}$$



3. Trigonometric ratios of an angle 3A in terms of angle A


  • sin3A = 3 sinA - 4sin3A


  • cos3A = 4 cos3A - 3 cosA


  • $$ \ tan3A = \frac{3 tanA - tan^3A}{1-3 tan^2A}$$


4. Trigonometric ratios of an angle 180

  • $$sin18^0 = \frac{-1+\sqrt{5}}{4}$$


  • $$cos18^0 = \frac{\sqrt{10+2\sqrt{5}}}{4}$$



5. Trigonometric ratios of an angle 360

  • $$ cos36^0 = \frac{1+\sqrt{5}}{4}$$


  • $$ sin36^0 = \frac{\sqrt{10-2\sqrt{5}}}{4}$$



6. Trigonometric ratios of an angle A in terms of angle A/2.

  • $$ sinA = 2 sin\frac{A}{2}cos\frac{A}{2}$$


  • $$ cosA = 1- 2 sin^2\frac{A}{2}$$


  • $$ tanA = \frac{2tan\frac{A}{2}}{1-tan^2\frac{A}{2}}$$


  • $$ sinA = \frac{2tan\frac{A}{2}}{1+tan^2\frac{A}{2}}$$


  • $$ cosA = \frac{1-tan^2\frac{A}{2}}{1+tan^2\frac{A}{2}}$$



7. Trigonometric ratios of an angle \(\frac{A}{2}\) in terms of cosA


  • $$sin\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{2}}$$


  • $$cos\frac{A}{2} = \pm \sqrt{\frac{1+cosA}{2}}$$


  • $$tan\frac{A}{2} = \pm \sqrt{\frac{1-cosA}{1+cosA}}$$



8. Trigonometric ratios of an angle \(\frac{A}{2}\) in terms of sinA



  • $$sin\frac{A}{2} + cos\frac{A}{2} = \pm \sqrt{1+sinA}$$


  • $$sin\frac{A}{2} - cos\frac{A}{2} = \pm \sqrt{1-sinA}$$



Series Expnsion of Trigonometric functions



  • sin θ = θ - θ3/3! + θ5/5! - θ7/7! .....

  • cos θ = 1 - θ2/2! + θ4/4! - θ6/6! .....

  • tan θ = θ + θ3/3 + 2θ5/15 .....


Approximate Value


If is θ small



  • sin θ ≈ θ

  • cos θ ≈ 1

  • tan θ ≈ θ


Average Value



  • < sin θ > = < sin nθ > = 0

  • < cos θ > = < cos nθ > = 0

  • < sin2 θ > = < sin2 nθ > = 1/2

  • < cos2 θ > = < cos2 nθ > = 1/2

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