formula of Trigonometry
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Answer:
By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:
sin θ = Opposite Side/Hypotenuse
cos θ = Adjacent Side/Hypotenuse
tan θ = Opposite Side/Adjacent Side
sec θ = Hypotenuse/Adjacent Side
cosec θ = Hypotenuse/Opposite Side
cot θ = Adjacent Side/Opposite Side
Reciprocal Identities
The Reciprocal Identities are given as:
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
All these are taken from a right angled triangle. When height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.
Periodicity Identities (in Radians)
These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.
sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
sin (π – A) = sin A & cos (π – A) = – cos A
sin (π + A) = – sin A & cos (π + A) = – cos A
sin (2π – A) = – sin A & cos (2π – A) = cos A
sin (2π + A) = sin A & cos (2π + A) = cos A
Co-function Identities (in Degrees)
The co-function or periodic identities can also be represented in degrees as:
sin(90°−x) = cos x
cos(90°−x) = sin x
tan(90°−x) = cot x
cot(90°−x) = tan x
sec(90°−x) = csc x
csc(90°−x) = sec x
Sum & Difference Identities
sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)
Double Angle Identities
sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
tan(2x) = [2tan(x)]/ [1−tan2(x)]
sec (2x) = sec2 x/(2-sec2 x)
csc (2x) = (sec x. csc x)/2
Triple Angle Identities
Sin 3x = 3sin x – 4sin3x
Cos 3x = 4cos3x-3cos x
Tan 3x = [3tanx-tan3x]/[1-3tan2x]
Half Angle Identities
sinx2=±1−cosx2−−−−−−√
cosx2=±1+cosx2−−−−−−√
tan(x2)=1−cos(x)1+cos(x)−−−−−−√
Also, tan(x2)=1−cos(x)1+cos(x)−−−−−−√=(1−cos(x))(1−cos(x))(1+cos(x))(1−cos(x))−−−−−−−−−−−−−√=(1−cos(x))21−cos2(x)−−−−−−−−√=(1−cos(x))2sin2(x)−−−−−−−−√=1−cos(x)sin(x) So, tan(x2)=1−cos(x)sin(x)
Product identities
sinx⋅cosy=sin(x+y)+sin(x−y)2
cosx⋅cosy=cos(x+y)+cos(x−y)2
sinx⋅siny=cos(x−y)−cos(x+y)2
Sum to Product Identities
sinx+siny=2sinx+y2cosx−y2
sinx−siny=2cosx+y2sinx−y2
cosx+cosy=2cosx+y2cosx−y2
cosx−cosy=−2sinx+y2sinx−y2
Inverse Trigonometry Formulas
sin-1 (–x) = – sin-1 x
cos-1 (–x) = π – cos-1 x
tan-1 (–x) = – tan-1 x
cosec-1 (–x) = – cosec-1 x
sec-1 (–x) = π – sec-1 x
cot-1 (–x) = π – cot-1 x
Answer:
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