त्रिकोणमितीय अनुपात

किसी समकोण त्रिभुज में किन्हीं दो भुजाओं के अनुपात को त्रिकोणमितीय अनुपात कहते है।

  • लम्ब/कर्ण = sin θ
  • आधार/कर्ण = cos θ
  • लम्ब/आधार = tan θ
  • आधार/लम्ब = cot θ
  • कर्ण/आधार = sec θ
  • कर्ण/लम्ब = cosec θ

त्रिकोणमितीय अनुपातों में परस्पर संबंध (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

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

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