# Trigonometry Formulas List

In Mathematics, trigonometry is the most important topic to learn. Trigonometry is basically the study of triangles where ‘Trigon’ means triangle and ‘metry’ means measurement. Also, trigonometry formulas list is created on the basis of trigonometry ratios such as sine, cosine and tangent. These formulas are used to solve various trigonometry problems.

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 ∞