Trigonometry Formulas & Identities (Complete List) (2024)

In Trigonometry, different types of problems can be solved using trigonometry formulas. These problems may includetrigonometric ratios (sin, cos, tan, sec, cosec and cot), Pythagorean identities, product identities, etc. Some formulas including the sign of ratios in different quadrants, involving co-function identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, etc., are also given in brief here.

Learning and memorizing these mathematics formulas in trigonometry will help the students of Classes 10, 11, and 12 to score good marks in this concept. They can find the trigonometry table along with inverse trigonometry formulas to solve the problems based on them.

Trigonometry Formulas PDF

Below is the link 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 are 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 List

When we learn about trigonometric formulas, we consider them for right-angled triangles only. In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Base). 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.

Here is the list of formulas for trigonometry.

• Basic Formulas
• Reciprocal Identities
• Trigonometry Table
• Periodic Identities
• Co-function Identities
• Sum and Difference Identities
• Double Angle Identities
• Triple Angle Identities
• Half Angle Identities
• Product Identities
• Sum to Product Identities
• Inverse Trigonometry Formulas

Basic Trigonometric Function Formulas

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

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

• sinθ=Opposite Side/Hypotenuse
• cos θ= Adjacent Side/Hypotenuse
• tan θ=Opposite Side/Adjacent Side
• sec θ=Hypotenuse/Adjacent Side
• cosecθ=Hypotenuse/Opposite Side
• cotθ=Adjacent Side/Opposite Side

Reciprocal Identities

TheReciprocal Identitiesare given as:

• cosecθ=1/sinθ
• secθ=1/cosθ
• cotθ=1/tanθ
• sinθ=1/cosecθ
• cosθ=1/secθ
• tanθ=1/cotθ

All these are taken from a right-angled triangle. When the height and base side of the right triangle are known, 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 Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Periodicity Identities (in Radians)

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A & cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

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

Cofunction Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x)=cosx
• cos(90°−x)=sinx
• tan(90°−x)=cotx
• cot(90°−x)=tanx
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

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)

• \(\begin{array}{l}tan(x+y)=\frac{tan~x+tan~y}{1-tan~x.tan~y}\end{array} \)

• sin(x–y)=sin(x)cos(y)–cos(x)sin(y)
• cos(x–y)=cos(x)cos(y)+sin(x)sin(y)

• \(\begin{array}{l}tan(x-y)=\frac{tan~x-tan~y}{1+tan~x.tan~y}\end{array} \)

Double Angle Identities

• \(\begin{array}{l}sin(2x) = 2sin~x.cos~x = \frac{2tan~x}{1+tan^{2}x}\end{array} \)

• \(\begin{array}{l}cos~2x = cos^{2}x – sin^{2}x = \frac{1-tan^{2}x}{1+tan^{2}x}\end{array} \)

• cos(2x)=2cos²(x)−1=1–2sin²(x)

• \(\begin{array}{l}tan~2x = \frac{2tan~x}{1-tan^{2}x}\end{array} \)

• \(\begin{array}{l}sec~2x = \frac{sec^{2}x}{2-sec^{2}x}\end{array} \)

• \(\begin{array}{l}cosec~2x = \frac{(sec~x.cosec~x)}{2}\end{array} \)

Triple Angle Identities

• Sin 3x = 3sin x – 4sin³x
• Cos 3x = 4cos³x-3cos x

• \(\begin{array}{l}Tan~3x = \frac{3~tan~x-tan^{3}x}{1-3tan^{2}x}\end{array} \)

Half Angle Identities

• \(\begin{array}{l}\sin\frac{x}{2}=\pm \sqrt{\frac{1-\cos\: x}{2}}\end{array} \)

• \(\begin{array}{l}\cos\frac{x}{2}=\pm \sqrt{\frac{1+\cos\: x}{2}}\end{array} \)

• \(\begin{array}{l}\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\end{array} \)

  See Also

  Explicit formulas for arithmetic sequences | Algebra (article) | Khan Academy

Also,

\(\begin{array}{l}\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))(1-\cos(x))}{(1+\cos(x))(1-\cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{1-\cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{\sin^{2}(x)}}\\ \\ \\ =\frac{1-\cos(x)}{\sin(x)}\end{array} \)

\(\begin{array}{l}\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}\end{array} \)

Product identities

• \(\begin{array}{l}\sin\: x\cdot \cos\:y=\frac{\sin(x+y)+\sin(x-y)}{2}\end{array} \)

• \(\begin{array}{l}\cos\: x\cdot \cos\:y=\frac{\cos(x+y)+\cos(x-y)}{2}\end{array} \)

• \(\begin{array}{l}\sin\: x\cdot \sin\:y=\frac{\cos(x-y)-\cos(x+y)}{2}\end{array} \)

Sum to Product Identities

• \(\begin{array}{l}\sin\: x+\sin\: y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\end{array} \)

• \(\begin{array}{l}\sin\: x-\sin\: y=2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\end{array} \)

• \(\begin{array}{l}\cos\: x+\cos\: y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\end{array} \)

• \(\begin{array}{l}\cos\: x-\cos\: y=-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\end{array} \)

Inverse Trigonometry Formulas

• sin^-1(–x) = – sin^-1x
• cos^-1(–x) = π – cos^-1x
• tan^-1(–x) = – tan^-1 x
• cosec^-1(–x) = – cosec^-1x
• sec^-1(–x) = π – sec^-1x
• cot^-1(–x) = π – cot^-1x

What is Sin 3x Formula?

Sin 3x is the sine of three times of an angle in a right-angled triangle, which is expressed as:

Sin 3x = 3sin x – 4sin³x

Trigonometry Formulas From Class 10 to Class 12

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 sides of the right triangle.

Here we provide a list of all Trigonometry formulas for the students. 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 if they appear in the examination.

We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. 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 download the trigonometric formulas pdf provided above.

Video Lesson on Trigonometry

Solved Problems

Q.1:

What is the value of (sin 30° + cos 30°) – (sin 60° + cos 60°)?

Solution:

Given,

(sin 30° + cos 30°) – (sin 60° + cos 60°)

= (½) + (√3/2) – (√3/2) – (½)

= 0

Q.2:

If cos A = 4/5, then find the value of tan A.

Solution:

Given,

cos A = ⅘

As we know, from trigonometry identities,

1 + tan²A = sec²A

sec²A – 1 = tan²A

(1/cos²A) -1 = tan²A

Putting the value of cos A = ⅘.

(5/4)² – 1 = tan² A

(25 – 16)/16 =tan² A

tan²A = 9/16

tan A = 3/4

Practice Problems

1. If sin A = 1/2 then find the value of sin 3A.
2. Prove that tanθ + tan(90° –θ) = secθ sec(90° –θ)
3. If tan A = 3/4, the find sin A cos A.