introduction to trigonometry
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Submitted By AMAL A S
INTRODUCTIONTO
TRIGONOMETRY
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The word trigonometry is derived from the ancient Greek language and means
measurement of triangles.
trigonon “triangle” +
metron “measure”=
Trigonometry
Trigonometry...?????
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a b
c
B A
C
Trigonometry ... is all about
Triangles…
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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
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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
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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
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Trigonometric functions..
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"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
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Their graphs as functions and the characteristics
•
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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
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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
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Proof for trigonometric ratios 30°,45°,60°
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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) .
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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°
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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°
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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°
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TRIGONOMETRIC IDENTITIES
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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
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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)
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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)
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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 ]
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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)]
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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