Submitted By AMAL A S

INTRODUCTIONTO

TRIGONOMETRY

The word trigonometry is derived from the ancient Greek language and means

measurement of triangles.

trigonon “triangle” +

metron “measure”=

Trigonometry

Trigonometry...?????

a b

c

B A

C

Trigonometry ... is all about 

Triangles…

A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side:

Adjacent is adjacent to the angle "θ“

Opposite is opposite the angle

The longest side is the Hypotenuse.

Right Angled Triangle

Hypotenuse

Opp

osit

e

Adjacent

θ

DEGREE MEASURE AND RADIAN MEASURE

O A

B

Terminal

Side

Initial Side

Degree measure: If a rotation from the initial side to terminal side is(1/360)th of a revolution, the angle is said to have a measure of one degree, written as 1°.

Degree measure= 180/ π x Radian measure

1

O

1

1

B

A

1

Radian measure: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have measure of 1 radian.

Radian measure= π/180 x Degree measure

ANGLES

Angles (such as the angle "θ" ) can be in Degrees or Radians. Here are some examples:

Angle Degree Radians

Right Angle 90° π/2

Straight Angle 180° π

Full Rotation 360° 2π

Trigonometric functions..

"Sine, Cosine and Tangent"The three most common functions in trigonometry are  Sine, Cosine and Tangent.

They are simply one side of a triangle divided by another.

For any angle "θ":

Sine Function: sin(θ) = Opposite / HypotenuseCosine Function: cos(θ) = Adjacent / HypotenuseTangent Function: tan(θ) = Opposite / Adjacent

Hypotenuse

Opp

osit

e

Adjacent

θ

Their graphs as functions and the characteristics

•  

TRGONOMETRIC FUNCTIONS

0° π/6

30°

π/4

45°

π/3

60°

π/2

90°

π

180°

3π/2

270°

2π

360°

sin 0 1/2 1/√2 √3/2 1 0 -1 0

cos 1 √3/2 1/√2 1/2 0 -1 0 1

tan 0 1/√3 1 √3 Not defined

0 Not defined

0

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Hypotenuse

Opp

osit

e

Adjacent

θ

Cosecant Function : csc(θ) = Hypotenuse / Opposite

Secant Function : sec(θ) = Hypotenuse / Adjacent

Cotangent Function : cot(θ) = Adjacent / Opposite

Proof for trigonometric ratios 30°,45°,60°

Computing unknown sides or angles in a right triangle.

In order to find a side of a right triangle you can use the Pythagorean Theorem, which is a2+b2=c2. The a and b represent the two shorter sides and the c represents the longest side which is the hypotenuse.

To get the angle of a right angle you can use sine, cosine, and tangent inverse. They are expressed as tan^(-1) ,cos^(-1) , and sin^(-1) .

o Find the sine, the cosine, and the tangent of 30°.

Begin by sketching a 30°-60°-90° triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Pythagoras Theorem , it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2.

sin 30° = opp./hyp. = 1/2 = 0.5

cos 30° = adj./hyp. = √3/2 ≈ 0.8660

tan 30° = opp./adj. = 1/√3 = √3/3 ≈ 0.5774

1

√3

2

30°

o Find the sine, the cosine, and the tangent of 45°.

Begin by sketching a 45°-45°-90° triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. The length of the hypotenuse is √2 (Pythagoras Theorem).

sin 45° = opp./hyp. = 1/√2 =2/√2≈ 0.7071

cos 45° = adj./hyp. = 1/√2 =2/√2≈ 0.7071

tan 45° = opp./adj. = 1/1 = 1

1

1

√2

45°

o Find the sine, the cosine, and the tangent of 60°.

Begin by sketching a 30°-60°-90° triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Pythagoras Theorem , it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2.

sin 60° = opp./hyp = √3/2 ≈ 0.8660

cos 60° = adj./hyp = ½ = 0.5

tan 60° = opp./adj. = √3/1 ≈ 1.7320

1

√3

2

30°

60°

TRIGONOMETRIC IDENTITIES

Reciprocal Identities

sin u = 1/csc u cos u = 1/sec u tan u = 1/cot u

csc u = 1/sin u

sec u = 1/cos u

cot u = 1/tan u

Pythagorean Identities

sin2 u + cos2 u = 1

1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

Quotient Identities

tan u = sin u /cos u

cot u =cos u /sin u

Co-Function Identities

sin( π/2− u) = cos u cos( π/2− u) = sin u

tan( π/2− u) = cot u cot( π/2− u) = tan u

csc( π/2− u) = sec u sec( π/2− u) = csc u

sin(−u) = −sin u cos(−u) = cos u tan(−u) = −tan u cot(−u) = −cot u csc(−u) = −csc u sec(−u) = sec u

Parity Identities (Even & Odd)

Sum & Di erence Formulas ff

sin(u ± v) = sin u cos v ± cos u sin v

cos(u ± v) = cos u cos v sin u sin v∓

tan(u ± v) = tan u ± tan v / 1 tan u tan v∓

Double Angle Formulas

sin(2u) = 2sin u cos u

cos(2u) = cos2 u − sin2 u = 2cos2 u − 1 = 1 − 2sin2 u

tan(2u) =2tanu /(1 − tan2 u)

Sum-to-Product Formulas

Sin u + sin v = 2sin [ (u + v) /2 ] cos [ (u − v ) /2 ]

Sin u − sin v = 2cos [ (u + v) /2 ] sin [ (u − v ) /2 ]

Cos u + cos v = 2cos [ (u + v) /2 ] cos [ (u − v) /2 ]

Cos u − cos v = −2sin [ (u + v) /2 ] sin [ (u − v) /2 ]

Product-to-Sum Formulas

Sin u sin v = ½ [cos(u − v) − cos(u + v)]

Cos u cos v = ½ [cos(u − v) + cos(u + v)]

Sin u cos v = ½ [sin(u + v) + sin(u − v)]

Cos u sin v = ½ [sin(u + v) − sin(u − v)]

sin2 u = 1 − cos(2u) / 2

cos2 u = 1 + cos(2u) / 2

tan2 u = 1 − cos(2u) / 1 + cos(2u)

Power – Reducing / Half Angle Formulas