Formula sheet: Trigonometric Functions

Trigonometric Functions16 min readLesson 1 of 7

1.1 Angles and arc length An angle measures the rotation from an initial ray to a terminal ray. Anticlockwise is positive and clockwise is negative. One full turn is 360 degrees or 2 pi radians, so the two units are linked by a fixed ratio. Half a turn is pi radians, a quarter turn is pi by 2 radians, and the standard conversions follow from those two facts. For a circle of radius r, take an arc of length l and the central angle theta that it cuts. The relation between them is l = r theta, where theta is measured in radians. This one rule handles every arc length and radius question, provided the angle is converted to radians first.

π rad = 18 0^{\circ}, 1^{\circ} = \frac{π}{180} rad, l = r θ

Degree and radian conversion, and the arc length relation

1.2 The six functions On the unit circle, a point reached after turning through x radians has coordinates (cos x, sin x). The other four functions are ratios built from these two: tan x is sin x over cos x, cot x is cos x over sin x, sec x is 1 over cos x, and cosec x is 1 over sin x. Division by zero is excluded from each domain. Three identities come from the unit circle and the ratios. The first is the Pythagorean identity for sine and cosine, and the other two divide it by cos squared x or by sin squared x.

sin^{2} x + cos^{2} x = 1, 1 + tan^{2} x = sec^{2} x, 1 + cot^{2} x = cosec^{2} x

The three fundamental identities

1.3 Signs, domains and ranges Function Domain Range sin x R [-1, 1] cos x R [-1, 1] tan x R without odd multiples of pi by 2 R cot x R without multiples of pi R sec x R without odd multiples of pi by 2 y at least 1 or y at most -1 cosec x R without multiples of pi y at least 1 or y at most -1 Signs follow the quadrant pattern. In the first quadrant all six are positive. In the second only sine and cosecant are positive. In the third tangent and cotangent are positive. In the fourth cosine and secant are positive.

Function	Domain	Range
sin x	R	[-1, 1]
cos x	R	[-1, 1]
tan x	R without odd multiples of pi by 2	R
cot x	R without multiples of pi	R
sec x	R without odd multiples of pi by 2	y at least 1 or y at most -1
cosec x	R without multiples of pi	y at least 1 or y at most -1

1.4 Sum and difference formulas These six lines build almost every other identity in the chapter. The cosine of a sum has a minus in the middle, while the cosine of a difference has a plus; the sine formulas keep the sign they are given. The tangent formulas follow on dividing sine by cosine, and they carry conditions that none of the angles is an odd multiple of pi by 2.

cos (x + y) = cos x cos y - sin x sin y, cos (x - y) = cos x cos y + sin x sin y

Cosine of a sum and of a difference

sin (x + y) = sin x cos y + cos x sin y, sin (x - y) = sin x cos y - cos x sin y

Sine of a sum and of a difference

1.5 Double and triple angle results Put y = x in the addition formulas and the double angle results drop out. The cosine of 2x has three equivalent forms; choose the one that matches the rest of the question. The triple angle formulas are used less often but appear directly in the exercises. Result Formula sin 2x 2 sin x cos x, also 2 tan x over 1 plus tan squared x cos 2x cos squared minus sin squared, also 2 cos squared minus 1, also 1 minus 2 sin squared tan 2x 2 tan x over 1 minus tan squared x sin 3x 3 sin x minus 4 sin cubed x cos 3x 4 cos cubed x minus 3 cos x tan 3x (3 tan x minus tan cubed x) over (1 minus 3 tan squared x)

Result	Formula
sin 2x	2 sin x cos x, also 2 tan x over 1 plus tan squared x
cos 2x	cos squared minus sin squared, also 2 cos squared minus 1, also 1 minus 2 sin squared
tan 2x	2 tan x over 1 minus tan squared x
sin 3x	3 sin x minus 4 sin cubed x
cos 3x	4 cos cubed x minus 3 cos x
tan 3x	(3 tan x minus tan cubed x) over (1 minus 3 tan squared x)

sin 2 x = 2 sin x cos x, cos 2 x = 1 - 2 sin^{2} x, tan 2 x = \frac{2 tan x}{1 - tan ^{2} x}

Double angle formulas

1.6 Transformation formulas The four sum to product formulas convert an addition of two sines or cosines into a product, which is the usual way to factor trigonometric expressions. The product to sum versions run the other way and are useful inside integrals later. A quick memory aid is that the angle outside the product is the average of the two angles, while the angle inside is half their difference. The minus sign in the cosine difference line is part of the formula, not a slip.

sin x + sin y = 2 sin \frac{x + y}{2} cos \frac{x - y}{2}, sin x - sin y = 2 cos \frac{x + y}{2} sin \frac{x - y}{2}

Sum to product formulas for sine

cos x + cos y = 2 cos \frac{x + y}{2} cos \frac{x - y}{2}, cos x - cos y = - 2 sin \frac{x + y}{2} sin \frac{x - y}{2}

Sum to product formulas for cosine

Exam

Write the condition with every tangent identity. The tangent of a sum is undefined at odd multiples of pi by 2, and an answer that ignores the condition loses a mark even when the algebra is right.

in hinglish

Formula sheet ke core points. Radian aur degree ka relation pi radian = 180 degree hai, aur degree ko radian me badalne ke liye pi/180 se multiply karo. Arc length ka formula l = r theta hai, jisme theta radian me hona chahiye. Unit circle par point ke coordinates (cos x, sin x) hote hain, aur teen fundamental identities yaad rakho: sin squared plus cos squared = 1, 1 plus tan squared = sec squared, 1 plus cot squared = cosec squared. Signs quadrant se aate hain: first me sab positive, second me sin aur cosec, third me tan aur cot, fourth me cos aur sec. Sum aur difference formulas me cos ke liye sign ulta hota hai, aur double angle me cos 2x ke teen forms yaad rakho. Transformation formulas me average aur half difference wala pattern hai. Tangent wale answers me condition likhna mat bhoolo.