Trigonometric or Circular Functions

What is Trigonometry?

Branch of mathematics that solves problems relating to plane and spherical triangles. Its principles are based on the fixed proportions of sides for a particular angle in a right triangle, the simplest of which are known as the sine(sin), cosine(cos), and tangent(tan). It is of practical importance in navigation, surveying, and simple harmonic motion in physics.
Invented by Hipparchus, trigonometry was developed by Ptolemy of Alexandria and was known to early Hindu and Arab mathematicians.

Sine

A function of an angle in a right-angled triangle which is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side).
Various properties in physics vary sinusoidally; that is, they can be represented diagrammatically by a sine wave (a graph obtained by plotting values of angles against the values of their sines). Examples include simple harmonic motion, such as the way alternating current (AC) electricity varies with time.

sin = opposite / hypotenuse or SOA

Cosine

A function of an angle in a right triangle found by dividing the length of the side adjacent to the angle by the length of the hypotenuse (the longest side). It is usually shortened to cos.
The two non-right angles of a right triangle add up to 90° and are, therefore, described as complementary angles (or co-angles). If the two non-right angles are a and b, it may be seen that sin a = cos b sin b = cos a. Therefore, the sine of each angle equals the cosine of its co- angle. For example, if the co-angles of a triangle are 30° and 60° sin 30° = cos 60° = 0.5 sin 60° = cos 30° = 0.8660.

cos = adjacent / hyotenuse or CAH

Tangent

A function of an acute angle in a right triangle, defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to it; a way of expressing the gradient of a line.

tan = opposite / adjacent or TOA

The ratio of sine, cosin & tangent can be summarized into a mnemonic : TOA CAH SOH, or
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Cotangent

A function of an angle in a right triangle found by dividing the length of the side adjacent to the angle by the length of the side opposite it. It is usually written as cotan, or cot and it is the reciprocal of the tangent of the angle, so that cot A = 1/tan A, where A is the angle in question.

Secant

The function of a given angle in a right triangle, obtained by dividing the length of the hypotenuse (the longest side) by the length of the side adjacent to the angle. It is the reciprocal of the cosine (sec = 1/cos).

Cosecant

A function of an angle in a right triangle found by dividing the length of the hypotenuse (the longest side) by the length of the side opposite the angle. Thus the cosecant of an angle A, usually shortened to cosec A, is always greater than (or equal to) 1. It is the reciprocal of the sine of the angle, that is, cosec A = 1/sin A.

Circular Function Formulas

Trigonometric Fundamental Relations and Identities
 Inverse Relations Division Relations Pythagorean Relations csc q = 1/sin q sec q = 1/cos q cot q = 1/tan q tan q = sin q/cos q cot q = cos q/sin q sin2 q + cos2 q = 1 1 + tan2 q = sec2q 1 + cot2 q = csc2q

Formulas for functions of two angles.
 Addition Substraction Double Angle Half of an Angle csc q = 1/sin q sec q = 1/cos q cot q = 1/tan q tan q = sin q/cos q cot q = cos q/sin q sin2 q + cos2 q = 1 1 + tan2 q = sec2q 1 + cot2 q = csc2q sin 1/2q = +-