INTRODUCTION
TO
TRIGONOMETRY
Submitted By
AMAL A S
Trigonometry...?????
The word trigonometry is derived from the
ancient Greek language and means
measurement of triangles.
...
Trigonometry ...
is all about
Triangles…

C

a

b
B

A

c
A right-angled triangle (the right angle is
shown by the little box in the corner) has
names for each side:
Adjacent is a...
DEGREE MEASURE AND RADIAN MEASURE
B

B

1
1
1
A

O
1

A

O

Initial Side
Degree measure: If a rotation from the
initial si...
ANGLES
Angles (such as the angle "θ" ) can be
in Degrees or Radians.
Here are some examples:
Angle

Degree

Radians

Right...
Trigonometric functions..
"Sine, Cosine and Tangent"
The three most common functions in trigonometry are
Sine, Cosine and Tangent.
They are simply...
Their graphs as functions and the
characteristics
•
TRGONOMETRIC FUNCTIONS

π/6

π/4

π/3

π/2

π

3π/2

2π

30°

0°

45°

60°

90°

180°

270°

360°

sin

0

1/2

1/√2

√3/2...
Other Functions (Cotangent, Secant, Cosecant)

Cosecant Function : csc(θ) = Hypotenuse / Opposite
Secant Function : sec(θ)...
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 Pytha...
o Find the sine, the cosine, and the tangent of 30 .
Begin by sketching a 30 -60 -90 triangle. To
make the calculations si...
o Find the sine, the cosine, and the tangent of 45 .
Begin by sketching a 45 -45 -90 triangle.
Because all such triangles ...
o Find the sine, the cosine, and the tangent of 60 .
Begin by sketching a 30 -60 -90 triangle. To
make the calculations si...
TRIGONOMETRIC IDENTITIES
 Reciprocal Identities

 Pythagorean Identities

sin u = 1/csc u

sin2 u + cos2 u = 1

cos u = 1/sec u

1 + tan2 u =...
 Co-Function Identities

 Parity Identities (Even & Odd)

sin( π/2− u) = cos u

sin(−u) = −sin u

cos( π/2− u) = sin ...
 Sum & Difference Formulas
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) = t...
 Sum-to-Product Formulas
Sin u + sin v = 2sin [ (u + v) /2 ] cos [ (u − v ) /2 ]
Sin u − sin v = 2cos [ (u + v) /2 ] si...
 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...
 Power – Reducing / Half Angle Formulas
sin2 u = 1 − cos(2u) / 2

cos2 u = 1 + cos(2u) / 2
tan2 u = 1 − cos(2u) / 1 + ...
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Introduction to trigonometry

  1. 1. INTRODUCTION TO TRIGONOMETRY Submitted By AMAL A S
  2. 2. Trigonometry...????? The word trigonometry is derived from the ancient Greek language and means measurement of triangles. trigonon “triangle” + metron “measure” = Trigonometry
  3. 3. Trigonometry ... is all about Triangles… C a b B A c
  4. 4. 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. Opposite Right Angled Triangle θ Adjacent
  5. 5. DEGREE MEASURE AND RADIAN MEASURE B B 1 1 1 A O 1 A O 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 . 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. Degree measure= 180/ π x Radian measure Radian measure= π/180 x Degree measure
  6. 6. 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
  7. 7. Trigonometric functions..
  8. 8. "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. Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent Opposite For any angle "θ": θ Adjacent
  9. 9. Their graphs as functions and the characteristics •
  10. 10. TRGONOMETRIC FUNCTIONS π/6 π/4 π/3 π/2 π 3π/2 2π 30° 0° 45° 60° 90° 180° 270° 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
  11. 11. Other Functions (Cotangent, Secant, Cosecant) Cosecant Function : csc(θ) = Hypotenuse / Opposite Secant Function : sec(θ) = Hypotenuse / Adjacent Cotangent Function : cot(θ) = Adjacent / Opposite Opposite  Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another: θ Adjacent
  12. 12. Proof for trigonometric ratios 30 ,45 ,60
  13. 13. 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) .
  14. 14. 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 2 1 30 tan 30 = opp./adj. = 1/√3 = √3/3 ≈ 0.5774 √3
  15. 15. 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 √2 1 45 tan 45 = opp./adj. = 1/1 = 1 1
  16. 16. 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 60 2 1 cos 60 = adj./hyp = ½ = 0.5 tan 60 = opp./adj. = √3/1 ≈ 1.7320 30 √3
  17. 17. TRIGONOMETRIC IDENTITIES
  18. 18.  Reciprocal Identities  Pythagorean Identities sin u = 1/csc u sin2 u + cos2 u = 1 cos u = 1/sec u 1 + tan2 u = sec2 u tan u = 1/cot u 1 + cot2 u = csc2 u csc u = 1/sin u  Quotient Identities sec u = 1/cos u tan u = sin u /cos u cot u = 1/tan u cot u =cos u /sin u
  19. 19.  Co-Function Identities  Parity Identities (Even & Odd) sin( π/2− u) = cos u sin(−u) = −sin u cos( π/2− u) = sin u cos(−u) = cos u tan( π/2− u) = cot u tan(−u) = −tan u cot( π/2− u) = tan u cot(−u) = −cot u csc( π/2− u) = sec u csc(−u) = −csc u sec( π/2− u) = csc u sec(−u) = sec u
  20. 20.  Sum & Difference Formulas 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)
  21. 21.  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 ]
  22. 22.  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)]
  23. 23.  Power – Reducing / Half Angle Formulas sin2 u = 1 − cos(2u) / 2 cos2 u = 1 + cos(2u) / 2 tan2 u = 1 − cos(2u) / 1 + cos(2u)

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