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Trigonometric Identities

by REX
Trigonometric Identities

The major trigonometry functions are sine, cosine, and tangent, while the other three functions are cot, sec, and cosec. All six trig functions are used to derive the trigonometric identities.

Trigonometric identities come in handy when trigonometric functions are used in an expression or equation. Trigonometric identities are true for every value of a variable that appears on both sides of an equation. Geometrically, these identities include one or more trigonometric functions (such as sine, cosine, and tangent).

What are Trigonometric Identities?

Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation.

There are several trigonometric identities relating to the angle and the length of the side of a triangle. Only the right-angle triangle has the trigonometric identities.

The six trigonometric ratios are the basics of all fundamental trigonometric identities. Sine, cosine, tangent, cosecant, secant, and cotangent are the arithmetic functions. All of these trigonometric ratios are defined using the sides of a right triangle, namely the adjacent, opposing, and hypotenuse sides.

Reciprocal Trigonometric Identities

We already know that cosecant, secant, and cotangent are the reciprocals of sin, cosine, and tangent.

As a result, the reciprocal identities are as follows:

  • sin θ = 1/cosecθ ; cosec θ = 1/sinθ
  • cos θ = 1/secθ ; sec θ = 1/cosθ
  • tan θ = 1/cotθ ; cot θ = 1/tanθ

Pythagorean Trigonometric Identities

The Pythagorean trigonometric identities are derived from Pythagoras’ theorem in trigonometry. When we apply Pythagoras’ theorem to the right-angled triangle shown below, we get:

=  +

is used to divide both sides.

/  + /  = /

sin2θ + cos2θ = 1

Complementary and Supplementary Trigonometric Identities

A complementary angle is a pair of two angles whose total is equal to 90°. θ is (90 – θ) is the complement of an angle. The complementary angle trigonometric ratios are as follows:

  • sin (90°- θ) = cos θ
  • cos (90°- θ) = sin θ
  • cosec (90°- θ) = sec θ

The supplementary angles are a pair of two angles whose total equals 180°. An angle’s supplement θ is (180 – θ). Supplementary angles are:

  • sin (180°- θ) = sinθ
  • cos (180°- θ) = -cos θ
  • cosec (180°- θ) = cosec θ

Trigonometry Formulas

Trigonometry formulae are collections of formulas that employ trigonometric identities to solve issues involving the sides and angles of a right-angled triangle. For certain angles, these trigonometry formulas contain trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. In the following sections, we will go through the formulae relating Pythagorean identities, product identities, cofunction identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, and so on.

List of Trigonometry Formulas

Trigonometry formulae are categorize into many groups based on the trigonometry identities used. Let’s have a look at the following sets of trigonometric formulae.

  • Formulas for Trig Ratios: These are trigonometry formulae for the basic trigonometric ratios sin, cos, tan, and so on.
  • Reciprocal Identities: Trigonometry formulae that deal with the reciprocal connection between trig ratios are include in this category.
  • Trigonometric Ratio Table: The trigonometry table depicts trigonometry values for standard angles.
  • Periodic Identities: These are trigonometry formulae that aid in determining the values of trig functions for a shift in angles by 2.
  • Trigonometry formulae for cofunction identities demonstrate the interrelationships between trigonometric functions.
  • Sum and Difference Identities: These trigonometry formulae are use to calculate the value of the trigonometric function for the sum or difference of angles.
  • Sum to Product Identities: These trigonometry formulae are use to express the sum or product of trigonometric functions.
  • Formulas for Inverse Trigonometry: Inverse trigonometry formulae contain formulas for inverse trig functions such as sine inverse, cosine inverse, and so on.


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