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Trigonometric Formulas

trigonometric Formulas

Also, read
Trigonometry for class 10
Applications of Trigonometry
Inverse Trigonometric Function

Trigonometric formulas:

When you enter in the chapter of trigonometry it seems vast one, but it is not what you believe. This chapter is so much easy to solve. The condition is if you have all the trigonometric formulas on your leaps. The trigonometric formulas include all trigonometric ratios, trigonometric identities, trigonometric sign rule, quadrant rule and some of the value of the trigonometric function of specific degrees.

Here, we listed the important trigonometric formulas:

 Basics:

1.Trigonometric Ratios:

Functions Value
sin A \(\frac{P}{H}\)
cos A \(\frac{B}{H}\)
tan A \(\frac{P}{B}\)
cosec A \(\frac{H}{P}\)
sec A \(\frac{H}{B}\)
cot A \(\frac{B}{P}\)
  1. Trigonometric formulas based on relations:

Functions Relations
tan A \(\frac{sin\:A}{cos\:A}\)
cot A \(\frac{cos\:A}{sin\:A}\)
sec A \(\frac{1}{cos\:A}\)
cosec A \(\frac{1}{sin\:A}\)
  1. Negative sign convention of Trigonometric function:

The negative sign (-A) has usual meaning in terms of (2π – A).

s.no. Function
1. sin(-A) = – sin A
2. cos(-A) = cos A
3. tan(-A) = – tan A
4. cosec(-A) = – cosec A
5. sec(-A) = sec A
6. cot(-A) = – cot A

Also, common used function in quadrants:

The sign convention in these quadrant is remembered as “All School To College

i.e.

  • All positive for 1st quadrant,
  • Sin and cosec positive for 2nd quadrant,
  • Tan and cot positive for 3rd quadrant,
  • Cos and sec positive in 4th .
S.no. Value
1. sin(90⁰-A) = cos A
2. cos(90⁰-A) = sin A
3. tan(90⁰-A) = cot A
4. cosec(90⁰-A) = sec A
5. sec(90⁰-A) = cosec A
6. cot(90⁰-A) = tan A

 

  1. Primary Trigonometric Identities:

s.no. Identities
1. sin2 A + cos2 A = 1
2. sec2 A – tan2 A = 1
3. cosec2 A – cot2 A = 1
  1. Trigonometric Formula As Product Rule:

s.no. Rules
1. sin (A + B) = sin A cos B + cos A sin B
2. sin (A – B) = sin A cos B – cos A sin B
3. cos (A + B) = cos A cos B – sin A sin B
4. cos (A – B) = cos A cos B + sin A sin B
5. tan (A + B) = \(\frac{tanA + tanB}{1 – tanA\: tanB}\)
6. tan (A – B) = \(\frac{tanA – tanB}{1 + tanA\: tanB}\)

 

  1. Trigonometric Formulas for Double Angle:

s.no. Formulae
1. sin 2A = 2 sin A∙cos A
2. cos 2A = cos2A – sin2A

=2cos2A – 1

= 1 – 2sin2A

3. tan 2A = \(\frac{2tanA}{1- {tan}^{2}A}\)
  1. Trigonometric Formulas for Triple Angle:

s.no. Formulae
1. Sin 3A = 3sinA – 4sin3A
2. Cos 3A = 4cos3A – 3cosA
3. Tan3A = \(\frac{3tanA – {tan}^{3}A}{1- 3{tan}^{2}A}\)
4. Cot3A = \(\frac{{cot}^{3}A – 3cotA}{3{cot}^{2}A – 1}\)
  1. Trigonometric product conversion:

s.no. Formulae
1. \(\frac{cos(A – B) – cos(A+B)}{2}\) = sinA.sinB
2. \(\frac{cos(A – B) + cos(A+B)}{2}\) = cosA.cosB
3. \(\frac{sin(A + B) + sin(A – B)}{2}\) = sinA.cosB
4. \(\frac{cos(A + B) + sin(A – B)}{2}\) = cosA.sinB
  1. Extended product rule:

s.no. Formulae
1. sinA.sinB = \(\frac{cos\frac{(A + B)}{2}.cos\frac{(A-B)}{2}}{2}\)
2. cosA.cosB = \(\frac{cos\frac{(A + B)}{2}.sin\frac{(A-B)}{2}}{2}\)
3. sinA.cosB = \(\frac{sin\frac{(A + B)}{2}.cos\frac{(A-B)}{2}}{2}\)
4. cosA.sinB = \(\frac{sin\frac{(A + B)}{2}.sin\frac{(A-B)}{2}}{2}\)
  1. Trigonometric Half Angle Identities:

s.no. Formulae
1. \(\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}\)
2. \(\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}\)
3. \(\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\)

 

4. \(\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\) Also, \(\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}\)

 

 

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