Trigonometric ratios and Trigonometric Functions
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
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:
- 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:
- 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:
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:
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:
Note: the sign + or – depends on the quadrant of the original angle.
Trigonometric Ratios of Complementary Angles
- Sine of a Complementary Angle:
- Cosine of a Complementary Angle:
- Tangent of a Complementary Angle:
Trigonometric Ratios of Supplementary Angles
- Sine of a Supplementary Angle:
- Cosine of a Supplementary Angle:
- Tangent of a Supplementary Angle:
Trigonometric Ratios of Conjugate Angles
- Sine of a Conjugate Angle:
- Cosine of a Conjugate Angle:
- Tangent of a Conjugate Angle:
Trigonometric Ratios of Negative Angles (Even-Odd Identities)
- Sine of a Negative Angle:
- Cosine of a Negative Angle:
- Tangent of a Negative Angle:
Trigonometric Ratios of Angles that Differs by 90°
- Sine of an Angle that Differs by 90º:
- Cosine of an Angle that Differs by 90º:
- Tangent of an Angle that Differs by 90º:
Trigonometric Ratios of Angles that Differs by 180°
- Sine of an Angle that Differs by 180º:
- Cosine of an Angle that Differs by 180º:
- Tangent of an Angle that Differs by 180º:
Angle Sum Identities
- Angle Sum Identity for Sine:
- Angle Sum Identity for Cosine:
- Angle Sum Identity for Tangent:
Angle Difference Identities
- Angle Difference Identity for Sine:
- Angle Difference Identity for Cosine:
- Angle Difference Identity for Tangent:
Double-Angle Formulas
- Double-Angle Identity for Sine:
- Double-Angle Identity for Cosine:
- Double-Angle Identity for Tangent:
Half-Angle Formulas
- Half-Angle Identity for Sine:
- Half-Angle Identity for Cosine:
- Half-Angle Identity for Tangent:
Triple-Angle Formulas
- Triple-Angle Identity for Sine:
- Triple-Angle Identity for Cosine:
- Triple-Angle Identity for Tangent:
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.
- Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1 or sec θ = 1/cos θ.
- 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.
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.
Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.