Trigonometry Formulas and Identities - Complete List

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Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. 

• Sine (sin): sin(θ) = opposite / hypotenuse
• Cosine (cos): cos(θ) = adjacent / hypotenuse
• Tangent (tan): tan(θ) = opposite / adjacent
• Cotangent (cot): cot(θ) = adjacent / opposite
• Secant (sec): sec(θ) = hypotenuse / adjacent
• Cosecant (csc): csc(θ) = hypotenuse / opposite

Example: In a right triangle, if the hypotenuse is 5 and the opposite side is 3, the sine of the angle opposite the 3 side would be sin(θ) = 3/5.

Reciprocal Identities:

• Reciprocal of sine: csc(θ) = 1 / sin(θ)
• Reciprocal of cosine: sec(θ) = 1 / cos(θ)
• Reciprocal of tangent: cot(θ) = 1 / tan(θ)

Example: If the sine of an angle is 0.6, the reciprocal of sine would be csc(θ) = 1/0.6 = 1.67

Trigonometry Table:

Trigonometry table is used to memorize the values of sine, cosine, and tangent for commonly used angles. Below is the table for trigonometry formulas for angles that are commonly used for solving problems :

Periodic Identities:

Periodic identities are useful when working with trigonometric functions because they allow you to simplify expressions by reducing the size of the angle in the argument.

• 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

This means that if you double the angle in the argument of the sine or cosine function, you can express it as a product of sine and cosine of that angle.

These identities are useful when solving trigonometric equations, proving trigonometric identities, and graphing trigonometric functions.

Co-function Identities:

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

Sum and Difference Identities:

• Sum of sine: sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
• Difference of sine: sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
• Sum of cosine: cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
• Difference of cosine: cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

Double Angle Identities:

• Sine of double angle: sin(2a) = 2sin(a)cos(a)
• Cosine of double angle: cos(2a) = cos^2(a) - sin^2(a)
• Tangent of double angle: tan(2a) = (2tan(a)) / (1 - tan^2(a))

Triple Angle Identities:

• Sine of triple angle: sin(3a) = 3sin(a) - 4sin^3(a)
• Cosine of triple angle: cos(3a) = 4cos^3(a) - 3cos(a)

Half Angle Identities:

• Sine of half angle: sin(a/2) = ± √[(1 - cos(a)) / 2]
• Cosine of half angle: cos(a/2) = ± √[(1 + cos(a)) / 2]
• Tangent of half angle: tan(a/2) = ± √[(1 - cos(a)) / (1 + cos(a))]

Product Identities:

• Sine of product: sin(a)sin(b) = (cos(a-b) - cos(a+b))/2
• Cosine of product: cos(a)cos(b) = (cos(a-b) + cos(a+b))/2

Sum to Product Identities:

• sin(a) + sin(b) = 2sin( (a+b) / 2)cos( (a-b) / 2)
• sin(a) - sin(b) = 2cos( (a+b) / 2)sin( (a-b) / 2)
• cos(a) + cos(b) = 2cos( (a+b) / 2)cos( (a-b) / 2)
• cos(a) - cos(b) = -2sin( (a+b) / 2)sin( (a-b) / 2)

Inverse Trigonometry Formulas:

• arcsine: sin^-1(x) = the angle whose sine is x
• arccosine: cos^-1(x) = the angle whose cosine is x
• arctangent: tan^-1(x) = the angle whose tangent is x
• arccotangent: cot^-1(x) = the angle whose cotangent is x
• arcsecant: sec^-1(x) = the angle whose secant is x
• arccosecant: csc^-1(x) = the angle whose cosecant is x

Example: If sin(a) = 0.8, then arcsin(0.8) = the angle whose sine is 0.8

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

Videos Lesson on Trigonometry

Trigonometry Formulas Major System:

Trigonometric Identities: Formulas that involve trigonometric functions and are true for all values of the variables.

Trigonometric Ratios: Formulas that express the relationship between the measurement of the angles and the length of the sides of a right triangle.

These formulas are helpful for 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.

Frequently Asked Questions on Trigonometry Formulas

What are the basic trigonometric ratios?

The basic trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. These ratios are used to relate the side lengths of a right-angled triangle to its angles.

What are the formulas for trigonometry ratios?

The formulas for trigonometry ratios are:

Sin A = Perpendicular/Hypotenuse

Cos A = Base/Hypotenuse

Tan A = Perpendicular/Base

What are the three main functions in trigonometry?

The three main functions in trigonometry are Sine, Cosine, and Tangent. These functions are used to describe the ratios of the side lengths of a right-angled triangle to its angles.

What are the fundamental trigonometry identities?

The fundamental trigonometry identities include:

•  sin2 A + cos2 A = 1 (Pythagorean identity)
• 1+tan2 A = sec2 A (reciprocal identity)
• 1+cot2 A = csc2 A (reciprocal identity)

What are trigonometric formulas used for?

Trigonometric formulas are used to evaluate problems that involve trigonometric functions such as sine, cosine, tangent, cotangent, cosecant, and secant. These formulas can be used to solve problems that involve the angles and sides of a right-angled triangle.

What is the formula for sin 3x?

The formula for sin 3x is 3sin x - 4sin³x. This is known as the triple angle identity.

Solved Problems

Solve for x in the equation: sin(x) = 0.6

Solution:

Using the trigonometry table, we know that sin(x) = 0.6 when x = 30 degrees or x = 150 degrees.

Solve for x in the equation: tan(x) = -2

Solution:

Using the trigonometry table, we know that tan(x) = -2 when x = -63.4 degrees or x = 246.6 degrees

Solve for x in the equation: cos(2x) = 0.8

Solution:

Using the double angle identity, cos(2x) = 2cos^2(x) - 1 = 0.8

so, cos^2(x) = 0.9

therefore, cos(x) = ± √0.9 = ± 0.948

Solve for x in the equation: 2sin(x)cos(x) = 0.4

Solution:

Using the double angle identity, 2sin(x)cos(x) = sin(2x) = 0.4

therefore, sin(2x) = 0.4

so, sin(x)cos(x) = 0.2

therefore, sin(x) = 0.2/cos(x)

Solve for x in the equation: cot(x) = 3

Solution:

Using the reciprocal identity, cot(x) = 1/tan(x) = 3

therefore, tan(x) = 1/3

so, x = tan^-1(1/3)

Solve for x in the equation: sec(x) = -4

Solution:

Using the reciprocal identity, sec(x) = 1/cos(x) = -4

therefore, cos(x) = -1/4

so, x = cos^-1(-1/4)

Solve for x in the equation: csc(x) = 5

Solution:

Using the reciprocal identity, csc(x) = 1/sin(x) = 5

therefore, sin(x) = 1/5

so, x = sin^-1(1/5)