Download Trigonometry

Trigonometry
Massoud Malek
Right-Angled Triangle
The sine of an angle is the ratio of the length of the opposite side to the length of the
hypotenuse.
opposite
a
sin A =
= .
hypothenuse
h
The cosine of an angle is the ratio of the length of the adjacent side to the length of the
hypotenuse.
adjacent
b
cos A =
= .
hypothenuse
h
The tangent of an angle is the ratio of the length of the opposite side to the length of the
adjacent side .
a
opposite
tan A =
= .
adjacent
b
Reciprocal Trigonometric functions.
The remaining three functions are best defined using the above three functions.
The cosecant is the reciprocal of sine:
csc A =
hypothenuse
h
= .
opposite
a
Massoud Malek
Trigonometry
The secant is the reciprocal of cosine:
csc A =
hypothenuse
h
= .
adjacent
b
The cotangent reciprocal of tangent:
cot A =
adjacent
b
= .
opposite
a
Unit Circle.
A unit circle is a circle of radius one centered at the origin (0, 0).
2
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Trigonometry
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Trigonometry
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x2 + y 2 = 1 =⇒ sin2 θ + cos2 θ = 1
By dividing both sides of the above identity by cos2 θ and sin2 θ respectively, we obtain:
sin2 θ cos2 θ
1
+
=
=⇒ tan2 θ + 1 = sec2 θ
2
2
cos θ cos θ
cos2 θ
1
sin2 θ cos2 θ
+
=
=⇒ 1 + cot2 θ = csc2 θ
2
2
2
sin θ sin θ
sin θ
Here are some other identities:
sin(a ± b) = sin a cos b ± sin b cos a
tan(a ± b) =
tan a ± tan b
1 ∓ tan a tan b
cos(a ± b) = cos a cos b ∓ sin a sin b
cot(a ± b) =
1 ± cot a cot b
cot a ∓ cot b
cos(a + b) + cos(a − b)
cos(a − b) − cos(a + b)
sin a sin b =
2
2
sin(a + b) + sin(a − b)
sin 2a
sin a cos b =
sin a cos a =
2
2
1 − cos 2a
1 + cos 2a
sin2 a =
cos2 a =
2
2
cos a cos b =
Limits.
sin x
=1
x→0 x
lim
sin x
=0
x→∞ x
lim
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Trigonometry
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Mac Lauren Series.
∞
X
x2n+1
sin x =
(−1)n
2n + 1
n=0
∞
X
x2n
cos x =
(−1)n
2n
n=0
Complex forms
sinx =
ei x − e−i x
2i
cos x =
ei x + e−i x
2
de Moivre’s formula.
(cos x + i sin x)n = cos n x + i sin n x
Laws of Sines and Cosines
Let T = (a, b, c; A, B, C) be a triangle , where a, b, c are the sides and A, B, C are the
angles. Then T satisfies the following conditions:
sin A
sin B
sin C
=
=
a
b
c
Law of Sines :
Law of Cosines : c2 = a2 + b2 − 2 a b cos C
Derivatives and Integrals.
R
f (x)
sin x
cos x
tan x
cot x
sec x
csc x
f 0 (x)
cos x
− sin x
sec2 x
− csc2 x
sec x tan x
− csc x cot x
f (x)dx − cos x
sin x
ln | sec x| ln | sin x| ln | sec x+tan x| ln | csc x−cot x|
Inverse Trigonometric Functions.
Let f (x) = sin x, then the inverse function
f −1 (x) = arcsin x = sin−1 x
is called the arcsine of x . If f (x) is a one to one function in a domain, then f −1 (x) is
also a function. Thus
sin(arcsin x) = x f or x ∈ [−1, 1]
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Trigonometry
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and
arcsin (sin x) = x
x ∈ [−π/2, π/2].
whenever
Similarly we may define arccos x = cos−1 x, arctan x, etc...
f (x)
sin−1 x
cos−1 x
tan−1 x
cot−1 x
sec−1 x
csc−1 x
Domain
[−1, 1]
[−1, 1]
(−∞, ∞)
(−∞, ∞)
|x| ≥ 1
|x| ≥ 1
π
2
(0, π)
[0, π] − { π2 }
1
1 + x2
−1
1 + x2
1
x x2 − 1
Range
f 0 (x)
|x| ≤
√
π
2
[0, π]
−1
√
1 − x2
1
1 − x2
|x| ≤
√
|x| ≤
π
2
−1
x x2 − 1
√
To integrate inverse trigonometric functions, we use integration by parts with v(x) = 1.
Z
Z
sin−1 x dx = x sin−1 x +
cos−1 x dx = x cos−1 x −
√
1 − x2 + C
√
1 − x2 + C
Z
tan−1 x dx = x tan−1 x −
1
ln(1 + x2 ) + C
2
Z
cot−1 x dx = x cot−1 x +
1
ln(1 + x2 ) + C
2
Z
Z
sec−1 x dx = x sec−1 x − ln |x +
csc−1 x dx = x csc−1 x + ln |x +
√
x2 − 1| + C
√
1 − x2 + C
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Trigonometry
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Identities.
sin−1 x + cos−1 x =
π
,
2
π
,
2
1
−1
−1
sin x = csc
,
x
sec−1 x + csc−1 x =
sec
−1
−1
x = cos
1
,
x
cos(sin−1 x) = sin(cos−1 x) =
sin−1 x = − sin−1 x
√
f or
|x| ≤ 1.
f or
|x| ≥ 1.
f or
|x| ≤ 1.
f or
|x| ≥ 1.
1 − x2
sec(tan−1 x) =
cos−1 x = π − cos−1 x
The following table is obtained from the unit circle:
√
x2 + 1
tan−1 x = − tan−1 x
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Trigonometry
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x
sin x
cos x
tan x
cot x
sec x
csc x
0
0
1
0
±∞
1
±∞
π/6
1/2
√
3/2
3/3
√
3
√
2 3/3
2
√
√
π/4
π/3
√
√
2/2
√
2/2
3/2
1/2
√
1
1
√
√
3/3
2
√
2 3/3
3
2
2
π/2
1
0
±∞
0
+∞
1
−x
− sin x
cos x
− tan x
− cot x
sec x
− csc x
π/2 − x
cos x
sin x
cot x
tan x
csc x
sec x
π/2 + x
cos x
− sin x
− cot x
− tan x
− csc x
sec x
π−x
sin x
− cos x − tan x
− cot x
− sec x
csc x
π+x
− sin x − cos x
tan x
cot x
− sec x − csc x
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Trigonometry
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The following table shows how trigonometric functions are related to each other.
f (x)
sin x
sin x
sin x
p
cos x
1−sin2 x
cos x
tan x
cot x
√
tan x
1
√
1−cos2 x √
2
1 + tan x
1 + cot2 x
cos x
√
1
1+tan2 x
√
sec x
csc x
sec2 x − 1
sec x
1
csc x
cot x
1
sec x
√
1+cot2 x
√
1−cos2 x
tan x p
cos x
1−sin2 x
tan x
1
cot x
√
p
1−sin2 x
cos x
√
cot x
sin x
1−cos2 x
1
tan x
cot x
√
p
1+tan2 x
cot x
√
1+cot2 x
sin x
1
sec x p
1−sin2 x
csc x
1
sin x
1
cos x
1
√
1−cos2 x
√
1+tan2 x
tan x
p
1+cot2 x
√
√
csc2 x−1
csc x
1
csc2 x−1
sec2 x−1
√
1
sec2 x−1
√
sec x
√
sec x
sec2 x−1
csc2 x−1
csc x
csc2 x−1
csc x