Trigonometry formulas are a collection that uses trigonometric identities to solve problems, involving the sides and angles of a right-angled triangle. For given angles, these trigonometry formulas include trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. While the trigonometric formulae involving trigonometric identities are the core of the subject, we also would like to understand the importance of trigonometric identities, which in a basic sense refers to an equation that involves trigonometric ratios of an angle.

In the following sections, trigonometric identities, including Pythagorean identities, product identities, co-function identities (shifting angles), sum & difference identities, double-angle identities, half-angle identities, and so on are explained in detail.

List of Trigonometric Formulas

When we first learn about trigonometric formulas, we only consider right-angled triangles. As we know, a right-angled triangle has three sides: the hypotenuse, the opposite side (perpendicular), and the adjacent side (Base). The longest side in a right-angled triangle is known as the hypotenuse, the opposite side is perpendicular, and the adjacent side is where both the hypotenuse and the opposite side rest. These sides and the basic structure of the right-angled triangle go a long way in determining the depth of understanding of trigonometry formulae. In short, the right-angled triangle is the reference point to derive or arrive at trigonometry formulae or trigonometric identities.

Here is a list of trigonometry formulas.

Basic Trigonometric Formulas

Inverse Trigonometric Formulas

Trigonometry Identities

Reciprocal Identities

Periodic Identities

Co-function Identities

Sum and Difference Identities

Double Angle Identities

Triple Angle Identities

Half Angle Identities

Product identities

Sum to Product Identities

Basic Trigonometric Formulas

In Trigonometry, there are six ratios that are utilized to find the elements. They are referred to as trigonometric functions. Sine, cosine, secant, cosecant, tangent, and cotangent are the six trigonometric functions.

Inverse Trigonometric Formulas

Trigonometric ratios are inverted using inverse trigonometry formulas to produce inverse trigonometric functions such as sin θ = x and θ=sin−1x. In this case, x can take the form of whole integers, decimals, fractions, or exponents.

Trigonometry Identities

Trigonometric Identities are equalities that involve trigonometry functions that stay valuable for all variables in the equation.

There are several trigonometric identities relating to the side length and angle of a triangle. These identities stay true to the right-angle triangle.

Reciprocal Identities

Trigonometric ratios feature reciprocal relation between a pair of ratios:

As explained, these are all derived from a right-angled triangle. If we know the height and base side of the right triangle, it will become easier to know sine, cosine, tangent, secant, cosecant, and cotangent values, by applying trigonometric formulas. We can also derive reciprocal trigonometric identities by applying trigonometric functions.

Periodicity Identities

The periodicity identities are formulas used to shift the angles by π/2, π, 2π, etc. If one observes keenly, fundamentally, all trigonometric identities are cyclic. They repeat after this periodicity constant. The periodicity constant varies among the trigonometric identities and is different for each.

Trigonometric Identities of Opposite Angles

As we dwell deep into trigonometry formulas and various other aspects of this branch of mathematics, we explore more interesting features that enhance our subject knowledge and take us through new paths of knowledge. One such is the trigonometric identities of opposite angles, where, a trigonometry angle that is measured in its clockwise direction, is measured in negative parity.

Complementary Angles Identities

As the expression suggests, complementary angles are the pair of angles whose added measure comes to 90°. Their trigonometric identities are:

Supplementary Angles Identities

These are a pair of angles whose measure adds to 180°. Their trigonometric identities are:

Periodicity of Trigonometric Function

Trigonometric functions, sin, cos, tan, cot, sec, and cosec, are all periodic and carry different periodicities. Their identities:

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