Trigonometry Formulas List
Trigonometry is basically the study of triangles where ‘Trigon’ means triangle and ‘metry’ means measurement.
Taking an example of the right angle triangle, trigonometry formulas list is made. All the trigonometric formulas are based on trigonometric identities and trigonometric ratios. The relationship between angles and length of the sides of the triangle is formulated with the help of trigonometry concepts.
Trigonometry formulas list will be helpful for students to solve trigonometric problems easily. Below is the list of formulas based on the right-angled triangle and unit circle which can be used as a reference to study trigonometry.
List of Important Trigonometry Formulas
First let us learn basic formulas of trigonometry, considering a right-angled triangle, which has an angle θ, a hypotenuse, a side opposite angle θ and a side adjacent to angle θ.
So the general trigonometry ratios for a right-angled triangle can be written as;
sinθ = \(\frac{Opposite side}{Hypotenuse}\)
cosθ = \(\frac{Adjacent Side}{Hypotenuse}\)
tanθ = \(\frac{Opposite side}{Adjacent Side}\)
secθ = \(\frac{Hypotenuse}{Adjacent side}\)
cosecθ = \(\frac{Hypotenuse}{Opposite side}\)
cotθ = \(\frac{Adjacent side}{Side opposite}\)
Similarly, for a unit circle, for which radius is 1, and θ is the angle.Then,
sinθ = y/1
cosθ = 1/y
tanθ = y/x
cotθ = x/y
secθ = 1/x
cosecθ = 1/y
Trigonometry Identities and Formulas
Tangent and Cotangent Identities
tanθ = \(\frac{sin\theta }{cos\theta }\)
cotθ = \(\frac{cos\theta }{sin\theta }\)
Reciprocal Identities
sinθ = 1/cosecθ
cosecθ = 1/sinθ
cosθ = 1/secθ
secθ = 1/cosθ
tanθ = 1/cotθ
cotθ = 1/tanθ
Pythagorean Identities
sin2θ + cos2θ = 1
1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ
Even and Odd Formulas
sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ) = -cosecθ
Cofunction Formulas
sin(900-θ) = cosθ
cos(900-θ) = sinθ
tan(900-θ) = cotθ
cot(900-θ) = tanθ
sec(900-θ) = cosecθ
cosec(900-θ) = secθ
Formulas for twice of angle
sin2θ = 2 sinθ cosθ
cos2θ = 1 – 2sin2θ
tan2θ = \(\frac{2tan\theta }{1-tan^2\theta }\)
Half Angle Formulas
sinθ = \(\pm \sqrt{\frac{1-cos2\theta }{2}}\)
cosθ = \(\pm \sqrt{\frac{1+cos2\theta }{2}}\)
tanθ = \(\pm \sqrt{\frac{1-cos2\theta }{1+cos2\theta}}\)
Formulas for Thrice of angle
sin3θ = 3sinθ – 4 sin3θ
Cos 3θ = 4cos3θ – 3 cosθ
Tan 3θ = \(\frac{3 tan\theta – tan^3\theta }{1-3tan^2\theta }\)
Cot 3θ = \(\frac{cot^3\theta – 3cot\theta }{3cot^2\theta-1 }\)
The Sum and Difference 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
Tan (A+B) = \(\frac{Tan A + Tan B}{1 – Tan A Tan B}\)
Tan (A-B) = \(\frac{Tan A – Tan B}{1 + Tan A Tan B}\)
The Product to Sum Formulas
Sin A Sin B = ½ [Cos (A-B) – Cos (A+B)]
Cos A Cos B = ½ [Cos (A-B) + Cos (A+B)]
Sin A Cos B = ½ [Sin (A+B) + Sin (A+B)]
Cos A Sin B = ½ [Sin (A+B) – Sin (A-B)]
The Sum to Product Formulas
Sin A + Sin B = 2 sin \(\frac{A+B}{2}\) cos \(\frac{A-B}{2}\)
Sin A – Sin B = 2 cos\(\frac{A+B}{2}\) sin \(\frac{A-B}{2}\)
Cos A + Cos B = 2 cos\(\frac{A+B}{2}\) cos \(\frac{A-B}{2}\)
Cos A – Cos B = – 2 sin\(\frac{A+B}{2}\) sin \(\frac{A-B}{2}\)
Inverse Trigonometric Functions
If Sin θ = x, then θ = sin-1 x = arcsin(x)
Similarly,
θ = cos-1x = arccos(x)
θ = tan-1 x = arctan(x)
Also, the inverse properties could be defined as;
sin-1(sin θ) = θ
cos-1(cos θ) = θ
tan-1(tan θ) = θ
Values for Trigonometry ratios:
|
Degrees |
00 |
300 |
450 |
600 |
900 |
1800 |
2700 |
3600 |
|
Radians |
0 |
π/6 |
π/4 |
π/3 |
π/2 |
π |
3π/2 |
2π |
|
Sinθ |
0 |
1/2 |
\(1/\sqrt{2}\) | \(\sqrt{3}/2\) |
1 |
0 |
-1 |
0 |
|
Cosθ |
1 |
\(\sqrt{3}/2\) | \(1/\sqrt{2}\) |
1/2 |
0 |
-1 |
0 |
1 |
|
Tanθ |
0 |
1/\(\sqrt{3}\) |
1 |
\(\sqrt{3}\) |
∞ |
0 |
∞ |
0 |
|
Cotθ |
∞ |
/\(\sqrt{3}\) |
1 |
1/\(\sqrt{3}\) |
0 |
∞ |
0 |
∞ |
|
Secθ |
1 |
2/\(\sqrt{3}\) |
/\(\sqrt{2}\) |
2 |
∞ |
-1 |
∞ |
1 |
|
Cosecθ |
∞ |
2 |
/\(\sqrt{2}\) |
2/\(\sqrt{3}\) |
1 |
∞ |
-1 |
∞ |