Trigonometric ratios and Trigonometric Functions

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Trigonometric ratios are defined as the ratios of the sides of a right triangle. There are six trigonometric ratios in total: sine, cosine, tangent, and their reciprocals, cosecant, secant and cotangent.

Trigonometric functions are real functions which relate an angle of a right triangle to ratios of two side lengths, with a defined range and domain. Each of these six trigonometric functions has a corresponding inverse function.

Trigonometric ratios

Drawing of the right triangle to calculate the sine

Trigonometric ratios are the ratios between the sides and angles of a right triangle. These ratios are given by the following trigonometric functions of the known angle θ, where a, b and c refer to the lengths of the right triangle’s sides.

Notice that we can label the sides, or legs, of the right triangle as opposite or adjacent, depending on the given angle. The hypotenuse c is always the side opposite to the right angle.

Let θ be one of the acute angles of the right triangle:

  • The sine is the ratio between the length of the oppisite leg (a) divided by the length of the hypotenuse (c). In a formula, it is written simply as sin:
    Sine formula
  • The cosine is the ratio between the length of the adjacent leg (b) divided by the length of the hypotenuse (c). In a formula, it is written simply as cos:
    Cosine formula
  • The tangent is the ratio between the length of the opposite leg (a) divided by the length of the adjacent leg (b). In a formula, it is written simply as tan:
    Tangent formula

Trigonometry Mnemonics. Remembering the Trigonometric Functions

The formulas of the sine, cosine and tangent ratios in a right triangle are collectively known by the mnemonic SohCahToa:

Sin = opposite / hypotenuse

Cos = adjacent / hypotenuse

Tan = opposite / adjacent

There are also other mnemonics like:

Some People Have……… Sin = Perpendicular / Hypotenuse

Curly Black Hair……… Cos = Base / Hypotenuse

Through Proper Brushing……… Tan = Perpendicular / Base

Trigonometric Ratios for Special Common Angles

The sine, cosine and tangent for the most common angles (0º, 30º, 45º, 60º, 90º, 180º and 270º) are:

Table of trigonometric ratios (cosecant, secant, cotangent) of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)

Relationships Between Trigonometric Ratios

Any trigonometric ratio can be expressed in terms of any other. The following table shows the formulas with which each one is expressed as a function of the other:

Table of the relationship between trigonometric ratios

Note: the sign + or – depends on the quadrant of the original angle.

Trigonometric Ratios of Complementary Angles

Trigonometric Ratios of Supplementary Angles

Trigonometric Ratios of Conjugate Angles

Trigonometric Ratios of Negative Angles (Even-Odd Identities)

Trigonometric Ratios of Angles that Differs by 90°

Trigonometric Ratios of Angles that Differs by 180°

Angle Sum Identities

Angle Difference Identities

Double-Angle Formulas

Half-Angle Formulas

Triple-Angle Formulas

Reciprocal Trigonometric Ratios

Reciprocal trigonometric ratios are the multiplicative inverses of trigonometric ratios. These are:

  • Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.
    Cosecant formula
  • Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1 or sec θ = 1/cos θ.
    Secant formula
  • Cotangent (cot): It is the reciprocal ratio of the tangent. Also in this case, cot θ · tan θ = 1 or cot θ = 1/tan θ.
    Cotangent formula

Inverse Trigonometric Functions

Inverse trigonometric functions are defined as the inverses of the trigonometric ratios. Namely, the inverse trigonometric functions are the inverse operation (for instance, multiplication and division are inverse operations) of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent).

The six basic inverse trigonometric functions are: arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent.

Trigonometric Functions

Trigonometric functions are also called circular functions. The reason is that in a right triangle that has been drawn on the coordinate axis, with the vertex of the angle θ at the center of a circle (O), that is, in standard position, its vertex B can go through all the points of this circle.

Drawing of the trigonometric functions of a triangle on a circle of radius 1

Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.


AUTHOR: Bernat Requena Serra

YEAR: 2021


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