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Trig cheat sheet - formulae
Introduction To Applied Mathematics (SAPM011)
University of Limpopo
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Definition of the Trig Functions
Right triangle definition
For this definition we assume that
0 < θ < π 2
or 0 ◦ < θ < 90 ◦.
sin(θ) = opposite hypotenuse
csc(θ) = hypotenuse opposite cos(θ) = adjacent hypotenuse
sec(θ) = hypotenuse adjacent
tan(θ) =
opposite adjacent cot(θ) =
adjacent opposite
Unit Circle Definition For this definition θ is any angle.
sin(θ) = y 1 =
y csc(θ) = 1 y cos(θ) = x 1 =
x sec(θ) = 1 x tan(θ) = xy cot(θ) = x y
Facts and Properties
Domain The domain is all the values of θ that can be plugged into the function.
sin(θ), θ can be any angle
cos(θ), θ can be any angle
tan(θ), θ 6 =
(
n + 1 2
)
π, n = 0, ± 1 , ± 2 ,...
csc(θ), θ 6 = nπ, n = 0, ± 1 , ± 2 ,...
sec(θ), θ 6 =
(
n + 1 2
)
π, n = 0, ± 1 , ± 2 ,...
cot(θ), θ 6 = nπ, n = 0, ± 1 , ± 2 ,...
Period The period of a function is the number, T , such that f (θ + T ) = f (θ). So, if ω is a fixed number and θ is any angle we have the following periods.
sin (ω θ) → T = 2
π ω cos (ω θ) → T = 2
π ω tan (ω θ) → T = π ω csc (ω θ) → T = 2 ωπ
sec (ω θ) → T = 2
π ω cot (ω θ) → T = π ω Range The range is all possible values to get out of the function.
− 1 ≤ sin(θ) ≤ 1 − 1 ≤ cos(θ) ≤ 1 −∞ < tan(θ) < ∞ −∞ < cot(θ) < ∞ sec(θ) ≥ 1 and sec(θ) ≤ − 1 csc(θ) ≥ 1 and csc(θ) ≤ − 1
Formulas and Identities
Tangent and Cotangent Identities
tan(θ) = sin(θ) cos(θ)
cot(θ) = cos(θ) sin(θ)
Reciprocal Identities
csc(θ) = sin 1 (θ) sin(θ) = csc 1 (θ)
sec(θ) =
1
cos(θ) cos(θ) =
1
sec(θ) cot(θ) =
1
tan(θ) tan(θ) =
1
cot(θ)
Pythagorean Identities
sin 2 (θ) + cos 2 (θ) = 1 tan 2 (θ) + 1 = sec 2 (θ) 1 + cot 2 (θ) = csc 2 (θ)
Even/Odd Formulas sin(−θ) = − sin(θ) csc(−θ) = − csc(θ) cos(−θ) = cos(θ) sec(−θ) = sec(θ) tan(−θ) = − tan(θ) cot(−θ) = − cot(θ)
Periodic Formulas
If n is an integer then,
sin(θ + 2πn) = sin(θ) csc(θ + 2πn) = csc(θ) cos(θ + 2πn) = cos(θ) sec(θ + 2πn) = sec(θ) tan(θ + πn) = tan(θ) cot(θ + πn) = cot(θ)
Degrees to Radians Formulas
If x is an angle in degrees and t is an angle in radians then π 180 =
t x
⇒ t = πx 180
and x = 180t π
Double Angle Formulas
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos 2 (θ) − sin 2 (θ)
= 2 cos 2 (θ) − 1 = 1 − 2 sin 2 (θ)
tan(2θ) =
2 tan(θ) 1 − tan 2 (θ)
Half Angle Formulas
sin
(
θ 2
)
= ±
√
1 − cos(θ) 2
cos
(
θ 2
)
= ±
√
1 + cos(θ) 2
tan
(
θ 2
)
= ±
√
1 − cos(θ) 1 + cos(θ)
Half Angle Formulas (alternate form) sin 2 (θ) = 12 (1 − cos(2θ)) cos 2 (θ) = 12 (1 + cos(2θ))
tan 2 (θ) = 1 − cos(2θ) 1 + cos(2θ)
Sum and Difference Formulas sin(α ± β) = sin(α) cos(β) ± cos(α) sin(β) cos(α ± β) = cos(α) cos(β) ∓ sin(α) sin(β)
tan(α ± β) =
tan(α) ± tan(β) 1 ∓ tan(α) tan(β) Product to Sum Formulas sin(α) sin(β) = 12 [cos(α − β) − cos(α + β)] cos(α) cos(β) = 12 [cos(α − β) + cos(α + β)] sin(α) cos(β) = 12 [sin(α + β) + sin(α − β)] cos(α) sin(β) = 12 [sin(α + β) − sin(α − β)]
Sum to Product Formulas sin(α) + sin(β) = 2 sin
(
α + β 2
)
cos
(
α − β 2
)
sin(α) − sin(β) = 2 cos
(
α + β 2
)
sin
(
α − β 2
)
cos(α) + cos(β) = 2 cos
(
α + β 2
)
cos
(
α − β 2
)
cos(α)−cos(β) = − 2 sin
(
α + β 2
)
sin
(
α − β 2
)
Cofunction Formulas sin
( π 2 − θ
)
= cos(θ) cos
( π 2 − θ
)
= sin(θ)
csc
( π 2
− θ
)
= sec(θ) sec
( π 2
− θ
)
= csc(θ)
tan
( π 2
− θ
)
= cot(θ) cot
( π 2
− θ
)
= tan(θ)
Inverse Trig Functions
Definition
y = sin− 1 (x) is equivalent to x = sin(y)
y = cos− 1 (x) is equivalent to x = cos(y)
y = tan− 1 (x) is equivalent to x = tan(y)
Domain and Range
Function Domain Range y = sin− 1 (x) − 1 ≤ x ≤ 1 − π 2
≤ y ≤ π 2 y = cos− 1 (x) − 1 ≤ x ≤ 1 0 ≤ y ≤ π y = tan− 1 (x) −∞ < x < ∞ −
π 2 < y < π 2
Inverse Properties cos
(
cos− 1 (x)
)
= x cos− 1 (cos(θ)) = θ sin
(
sin− 1 (x)
)
= x sin− 1 (sin(θ)) = θ tan
(
tan− 1 (x)
)
= x tan− 1 (tan(θ)) = θ
Alternate Notation sin− 1 (x) = arcsin(x) cos− 1 (x) = arccos(x) tan− 1 (x) = arctan(x)
Law of Sines, Cosines and Tangents
Law of Sines sin(α) a
= sin(β) b
= sin(γ) c
Law of Cosines
a 2 = b 2 + c 2 − 2 bc cos(α)
b 2 = a 2 + c 2 − 2 ac cos(β)
c 2 = a 2 + b 2 − 2 ab cos(γ)
Mollweide’s Formula a + b c =
cos
( 1
2 (α − β)
)
sin
( 1
2 γ
)
Law of Tangents a − b a + b
=
tan
( 1
2 (α − β)
)
tan
( 1
2 (α + β)
)
b − c b + c =
tan
( 1
2 (β − γ)
)
tan
( 1
2 (β + γ)
)
a − c a + c
=
tan
( 1
2 (α − γ)
)
tan
( 1
2 (α + γ)
)
Trig cheat sheet - formulae
Course: Introduction To Applied Mathematics (SAPM011)
University: University of Limpopo
