Trigonometric Functions Formulas - Single,Half,Double,Multiple Angles

Basic Trigonometric Functions

Definition of Trigonometric Functions For a Right Angle

Triangle ABC bas a right angle (9O°) at C and sides of length a, b, c. The trigonometric functions of angle A are defined as follows.

Extensions To Angles Which May be Greater Than 90°

Consider an xy coordinate system . A point P in the xy plane has coordinates (x,y) where x is eonsidered as positive along OX and negative along OX´ while y is positive along OY and negative along OY´ .

The distance from origin 0 to point P is positive and denoted by r = √x² + y²

The angle A described counterclockwise from OX is considered positive. If it is described dockhse from

OX it is considered negative. We cal1 X´ OX and Y´ OY the x and y axis respectively.

The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quadrants respectively

For an angle A in any quadrant the trigonometric functions of A are defined as follows.

sin A = y/r

cos A = x/r

tan A = y/x

cet A = x/y

sec A = r/x

csc A = r/y

Signs And Variations Of Trigonometric Functions

tan A = Sin A / Cos A

Cot A = 1 / tan A = Cos A / Sin A

Sec A = 1 / Cos A

Csc A = 1 / Sin A

Sin² A + Cos² A = 1

Sec² A - tan² A = 1

Csc² A - Cot² A = 1

Exact Values For Trigonometric Functions Of Various Angles

Functions Of Negative Angles

Sin (-A) = - Sin A

Cos (-A) = Cos A

tan (-A) = - tan A

Csc (-A) = - Csc A

Sec (-A) = Sec A

Cot (-A) = - Cot A

Addition Formulas

Functions Of Angles in All Quadrants in Terms Of Those in Quadrant I

Relationships Amoung Functions Of Angles in Quadrant I

Double Angle Formulas

Sin 2A = 2 Sin A Cos A

Cos 2A = Cos²A - Sin²A = 1 - 2 Sin²A = 2 Cos²A - 1

tan 2A = 2 tan A / 1 - tan²A