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Important Trigonometric Formulas in Mathmatics

Trigonometric Formulas: Are you searching for any trigonometric formula to solve any trigonometric question? Here we have covered all most all possible formulas with fully explanation.

trigonometric formulas

You may be preparing yourself for any competitive examinations or for your college or school examinations. Trigonometric is one subject which will help you fetch good marks in examinations. Because in the trigonometric section you just have to put the formula and get the answer.  It’s a tricky subject. So it’s very important for you to memorize all the trigonometric formulas.

tringle in trigonometric formulas

In the above figure let us assume that ∠ABC=θ (Where 0< θ<π/2)

Then the six trigonometric ratios are: sine, cosine, tangent, cotangent, cosecant for θ, and which can be written as follows:

  • sinθ
  • cosθ
  • tanθ
  • cotθ
  • secθ
  • cosecθ

As per the figure we have m∠ACB= 90° and the lengths

AC=p (perpendicular)

BC=b (base)

AC=h (hypotenuse)

Trigonometric Ratios Formula

From the above picture we can derive the following Trigonometric formulas as follows:

  • sinθ=\frac{AC}{AB}=\frac{a}{b}
  • cosθ=\frac{BC}{AB}=\frac{b}{h}
  • tanθ=\frac{AC}{BC}=\frac{p}{b}
  • cosecθ=\frac{AB}{AC}=\frac{h}{p}
  • secθ=\frac{AB}{BC}=\frac{h}{b}
  • cotθ=\frac{BC}{AC}=\frac{b}{p}

Reciprocal Formulas

  • sinθ=\frac{1}{cosec\Theta }
  • cosθ=\frac{1}{sec\Theta }
  • tanθ=\frac{1}{cot\Theta }
  • cosecθ= \frac{1}{sin\Theta }
  • secθ=\frac{1}{cos\Theta }
  • cotθ=\frac{1}{tan\Theta }
  • sinθ×cosecθ=1
  • cosθ×secθ=1
  • tanθ×cotθ=1
  • tanθ=\frac{sin\Theta }{cos\Theta }
  • cotθ=\frac{cos\Theta }{sin\Theta }

Pythagorean Identities Trigonometric Formulas

  • sin²θ+cos²θ=1
  • sec²θ-tan²θ=1
  • cosec²θ-cot²θ=1
  •  cosec²θ=1+cot²θ
  • cot²θ=cosec²θ-1
  • 1-sin²θ=cos²θ
  • 1-cos²θ=sin²θ
  • 1+tan²θ=sec²θ
  • tan²θ=sec²θ-1

Trigonometric Ratios of Combained Angles

  • sin(A+B)= sinA. cosB + cosA. sinB
  • sin(A-B)= sinA.cosB – cosA.sinB
  • cos(A+B)= cosA.cosB – sinA.sinB
  • cos(A-B)= cosA.cosB + sinA.sinB
  • tan(A+B)=\frac{tanA+tanB}{1-tanA.tanB}
  • tan(A-B)=\frac{tanA-tanB}{1+tanA.tanB}
  • cot(A+B)= \frac{cotA.cotB-1}{cotA+cotB}
  • cot(A-B)=\frac{cotA.cotB+1}{cotB-cotA}
  • sin(A+B) sin(A-B)=sin²A-sin²B=cos²B-cos²A
  • cos(A+B). cos(A-B)=cos²A-sin²B=cos²B – sin²A
  • sin(A+B+C)= sinA.cosB.cosC + cosA.cosB.sinC + cosA.sinB.cosC – sinA.sinB.sinC
  • cos(A+B+C)= cos A.cos B cos C – sin A sin B sin C – sin C.sin A.cos B – sin B.sin C.cos A
  • tan(A+B+C)=\frac{tan A + tan B + tan C - tan A tan B tan C}{ 1 - tan A tan B- tan C tan A - tan B tan C}

Product into Sum and Difference

  • 2 sinA.cosB = sin(A + B) + sin(A – B)
  • cosA.sinB = sin(A + B) – sin(A – B)
  • 2 sinA.sinB=cos(A-B) – cos(A+B)
  • cosA.cosB = cos(A + B) + cos(A – B)

Other Important formula

Here we have updated some other important Trigonometric formulas:

  • 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{C-D}{2}
  • sin2A=2sinA.cosB=\frac{2tanA}{1+tan^{2}A}
  • cos2A=cos²A-sin²A=1-2sin²A=2cos²A-1=\frac{1-tan^{2}A}{1+tan^{2}A}=\frac{cotA-tanA}{cotA+ tanA}
  • cos²A=½(1+cos2A)
  • sin²A=½(1-cos2A)
  • tan2A=tanA(1 +sec2A) = \frac{2tan}{1-tan^{2}A}
  • sin3A= 3sinA-4sin³A
  • tan3A= \frac{3tanA-tan^{3}A}{1-3tan^{2}A}
  • cot2A=cotA-cosec2A= (cotA – tanA)/2\frac{cot^{2}A-1}{2cotA}
  • cos3A=4cos³A – 3cos²A
  • cot 3A= \frac{3cotA-cot^{3}A}{1-3cot^{2}A}

Trigonometric Ratio Values

Angle

Sin

Cos

tan

cosec

sec

cot

Degree

Radian

0

0

1

0

1

30°

\frac{\pi }{6}\frac{1}{2}\frac{\sqrt{3}}{2}\frac{1}{\sqrt{3}}

2

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

√3

45°

\frac{\pi }{4}\frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}

1

√2

√2

1

60°

\frac{\pi }{3}\frac{\sqrt{3}}{2}\frac{1}{2}

√3

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

2

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

90°

\frac{\pi }{2}

1

0

1

0

120°

\frac{2\pi }{3}\frac{\sqrt{3}}{2}

\frac{1}{2}

-√3

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

-2

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

135°

\frac{3\pi }{4}\frac{1}{\sqrt{2}}

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

-1

√2

-√2

-1

150°

\frac{5\pi }{6}\frac{1}{2}

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

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

2

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

-√3

180°

\pi

0

-1

0

-1

210°

\frac{7\pi }{6}\frac{1}{2}\frac{\sqrt{3}}{2}\frac{1}{\sqrt{3}}-2\frac{2}{\sqrt{3}}√3

225°

\frac{5\pi }{4}\frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}1-√2-√21

240°

\frac{4\pi }{3}\frac{\sqrt{3}}{2}\frac{1}{2}√3\frac{2}{\sqrt{3}}-2\frac{1}{\sqrt{3}}

270°

\frac{3\pi }{2}

-1

0

-1

0

300°

\frac{5\pi }{3}

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

\frac{1}{2}

-√3

-√3

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

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

315°

\frac{7\pi }{4}

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

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

-1

-√2

√2

-1

330°

\frac{11\pi }{6}

\frac{1}{2}

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

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

-2

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

-√3

360°

 2π

0

1

1

Some Important Trigonometric Angle Values

  • sin15°=(√3-1)/2√2
  • cos15°=(√3+1)/2√2
  • tan15°=2-√3
  • sin18°=(√5-1)/4
  • cos18°=√(10+2√5)/4
  • sin22½°= √(2-√2)/2
  • cos22½°=√(2-√2)/2
  • tan22½°= √2-1
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