# Trigonometry Formulas

Trigonometry formulas provided below can help students get acquainted with different formulas, which can be helpful in solving questions on trigonometric with ease. Trigonometry problems are diverse and learning the below formulae will help in solving them better. Multiple formulae could be required to solve the problem, so learn and practice each formula to understand where to use them. Below is the link is given to download the pdf format of Trigonometry formulas for free so that students can learn them offline too.

Trigonometry is a branch of mathematics that deals with triangles. Trigonometry is also known as the study of relationships between lengths and angles of triangles.

There is an enormous number of uses of trigonometry and its formulae. For example, the technique of triangulation is used in Geography to measure the distance between landmarks; in Astronomy, to measure the distance to nearby stars and also in satellite navigation systems.

## Trigonometry Formulas Major systems

All trigonometric formulas are divided into two major systems:

• Trigonometric Identities
• Trigonometric Ratios

Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle.

Here we provide the students with a list of all Trigonometry formula. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, which are sometimes asked in the examination.

We also provide the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard value. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.

## Trigonometry Functions Formulas

In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular) and Adjacent side (Height). The longest side is known as the hypotenuse, the side opposite to the angle is perpendicular and the side where both hypotenuse and opposite side rests is the adjacent side.

There are basically 6 Laws used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions or identities are derived:

• sin θ = Opposite Side/Hypotenuse
• sec θ = Hypotenuse/Adjacent Side
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

The Reciprocal Identities are given as:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

All these are taken from a right angled triangle. With the length and base side of the right triangle given, we can find out the sine, cosine, tangent, secant, cosecant and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

## Trigonometry Formulas List

### A.Trigonometry Formulas involving Periodicity Identities (in Radians)

• sin(x+2πn) = sin x
• cos(x+2πn) = cos x
• tan(x+πn) = tan x
• cot(x+πn) = cot x
• sec(x+2πn) = sec x
• csc(x+2πn) = csc x

where n is an integer.

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identity. tan 45 = tan 225 but this is true for cos 45 and cos 225. Refer to the above trigonometry table to verify the values.

### B.Trigonometry Formulas involving Cofunction Identities (in Degrees)

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = csc x
• csc(90°−x) = sec x

### C.Trigonometry Formulas involving Sum/Difference Identities:

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

### D.Trigonometry Formulas involving Double Angle Identities:

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
• tan(2x) = [2tan(x)]/ [1−tan2(x)]
• sec (2x) = secx/(2-sec2 x)
• csc (2x) = (sec x. csc x)/2

### E.Trigonometry Formulas involving Half Angle Identities:

• $\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}$
• $\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}$
• $\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$

Also, $\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))(1-\cos(x))}{(1+\cos(x))(1-\cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{1-\cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{\sin^{2}(x)}}\\ \\ \\ =\frac{1-\cos(x)}{\sin(x)}$ So, $\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}$

### F.Trigonometry Formulas involving Product identities:

• $\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}$
• $\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}$
• $\sin\: x\cdot \sin\:y=\frac{\cos(x+y)-\cos(x-y)}{2}$

### G.Trigonometry Formulas involving Sum to Product Identities:

• $\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$
• $\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$
• $\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$
• $\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$