What are the different trigonometric formulae?

Answer 1

In a right angled triangle with sides Adjacent (the angle is between this side and the hypotenuse), Opposite (this side is the side opposite the angle) and Hypotenuse (the side opposite the right angle).

The six commonly used trigonometric ratios are:

sine = opposite / hypotenuse

cosine = adjacent / hypotenuse

tangent = opposite / adjacent = sine / cosine

cotangent = 1/tangent = adjacent / opposite = cosine / sine

secant = 1/cosine = hypotenuse / adjacent

cosecant = 1/sine = hypotenuse / opposite

There are various mnemonics to remember the first three of these ratios. Two such mnemonics which use the initial letters are:

1: A nonsense word:

SOHCAHTOA (pronounced sock-a-toe-ah or soh-ka-toe-ah)

S = O/H

C = A/H

T = O/A

2: A little rhyme:

Two Old Arabs

Soft Of Heart

Coshed Andy Hatchett

T = O/A

S = O/H

C = A/H

The trigonometric functions are periodic:

• Sine (sin):

Starts at 0° with a value of 0. It increases until it reaches 1 at 90°; then it decreases, reaching 0 again at 180° and continues onto -1 at 270°. Then it increases again, reaching 0 at 360° where it starts to repeat.

• Cosine (cos):

Starts at 0° with a value of 1. It decreases, reaching 0 at 90° and continues onto -1 at 180°. Then it increases, reaching 0 at 270° and continues onto 1 at 360° where it starts to repeat.

• Tangent (tan):

Starts at 0° with a value of 1. It increases towards an asymptote at 90°; it continues increasing, but from the negative side until it reaches 0 at 180° where it starts to repeat.

• Cosecant (csc = 1/sin):

Starts with an asymptote at 0° and decreases from the positive side until it reaches 1 at 90°; where it then increases towards another asymptote at 180°; then it continues increasing from the negative side until it reaches -1 at 270° before decreasing again towards the asymptote at 360° where it starts to repeat.

• Secant (sec = 1/cos):

Starts at 1 and increases towards an asymptote at 90°; it then increases from the negative side until it reaches -1 at 180° before decreasing again towards another asymptote at 270°. The it decreases fro the positive side until it reaches 1 at 360° and starts to repeat.

• Cotangent (cot = 1/tan):

Starts with an asymptote at 0° and decreases towards 0 at 90°; it then continues to decrease towards another asymptote at 180° where it starts to repeat.

As a result of this periodic nature, they have specific signs in the different quadrants of the cartesian plane:

• Sine: positive: I, II; negative: III, IV
• Cosine: positive: I, IV; negative: II, III
• Tangent: positive: I, III; negative: II, IV
• Cosecant: positive: I, II; negative: III, IV
• Secant: positive: I, IV; negative: II, III
• Cotangent: positive: I, III; negative: II, IV

If the angle is measured in radians, then the slopes of the trigonometric functions can be found by differentiating the functions:

d/dx sin x = cos x

d/dx cos x = -sin x

d/dx tan x = sec² x

d/dx csc x = -csc x cot x

d/dx sec x = sec x tan x

d/dx cot x = -csc² x