Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition
Unit Circle Definition Assume θ can be any angle.
Assume that: 0 < θ < π2 or 0◦ < θ < 90 ◦
y
(x, y ) 1
hypotenuse opposite
y
θ
x
x θ adjacent
sin θ =
opp hyp
adj adj cos θ = hyp
tan θ =
opp adj adj
hyp opp
sin θ =
hyp sec θ = adj adj
cos θ =
adj adj opp
tan θ =
csc θ =
cot θ =
y
csc θ =
1 x
sec θ =
1
1 y
1
x x cot θ = y
y x
Domains of the Trig Functions sin θ, cos θ, tan θ,
∀ θ ∈ (−∞, ∞) ∀ θ ∈ (−∞, ∞) 1 π, where n 2
∀ θ = n +
∈ Z
csc θ,
∀ θ = nπ, where n ∈ Z
sec θ,
∀
cot θ,
θ = n +
1 π, where n 2
nπ, where where n ∈ Z ∀ θ = nπ,
Ranges of the Trig Functions
−1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 −∞ ≤ tan θ ≤ ∞
csc θ sec θ
≥ 1 and csc θ ≤ −1 ≥ 1 and sec θ ≤ −1 −∞ ≤ cot θ ≤ ∞
Periods of the Trig Functions 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(ωθ ) cos(ωθ )
⇒
⇒ tan(ωθ ) ⇒
T =
2π
ω 2π T = ω π T = ω
csc(ωθ ) sec(ωθ )
⇒
⇒ cot(ωθ ) ⇒
T =
2π
ω 2π T = ω π T = ω
∈Z
Identities and Formulas Tangent and Cotangent Identities
sin θ tan θ = cos θ
Half Angle Formulas
cos θ cot θ = sin θ
1 csc θ 1 cos θ = sec θ 1 tan θ = cot θ
csc θ =
tan θ =
± β ) = sin α cos β ± cos α sin β cos(α ± β ) = cos α cos β ∓ sin α sin β tan α ± tan β tan(α ± β ) = 1 ∓ tan α tan β
tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ
Product to Sum Formulas
Even and Odd Formulas
sin( θ) = sin θ cos( θ) = cos θ tan( θ) = tan θ
csc( θ) = csc θ sec( θ) = sec θ cot( θ) = cot θ
− − − − −
− − −
− −
If n is an integer sin(θ + 2 πn ) = sin θ cos(θ + 2 πn) = cos θ tan(θ + πn ) = tan θ
1 sin α sin β = [cos(α β ) cos(α + β )] 2 1 cos α cos β = [cos(α β ) + cos(α + β )] 2 1 sin α cos β = [sin(α + β ) + sin(α β )] 2 1 cos α sin β = [sin(α + β ) sin(α β )] 2
− − −
−
Periodic Formulas
csc(θ + 2 πn) = csc θ sec(θ + 2 πn) = sec θ cot(θ + πn) = cot θ
−
−
cos(2θ) = cos2 θ sin2 θ = 2 cos2 θ 1 = 1 2sin2 θ
− −
−
2tan θ 1 tan2 θ
− − − −
α + β
cos
α
β
−
Cofunction Formulas
−
Degrees to Radians Formulas sin If x is an angle in degrees and t is an angle in
radians then:
csc
⇒
2 2 α + β α β sin α sin β = 2 cos sin 2 2 α + β α β cos α + cos β = 2 cos cos 2 2 α + β α β cos α cos β = 2sin sin 2 2
sin(2θ) = 2 sin θ cos θ
t = 180◦ x
−
Sum to Product Formulas
sin α + sin β = 2 sin
Double Angle Formulas
π
1 cos(2θ) 1 + cos(2θ)
sin(α
sin2 θ + cos2 θ = 1
tan(2θ) =
±
Sum and Difference Formulas
Pythagorean Identities
−
cos(2θ) 2
1 + cos(2θ) 2
cos θ =
1 sin θ 1 sec θ = cos θ 1 cot θ = tan θ
sin θ =
1
sin θ =
Reciprocal Identities
− ± ± −
t =
πx
and x = 180◦
180◦ t π
π
− − −
tan
2
π
2
π
2
θ = cos θ
cos
θ = sec θ
sec
θ = cot θ
cot
π
− − − 2
π
2
π
2
θ = sin θ
θ = csc θ θ = tan θ
Unit Circle (0, 1)
(
−
(
−
(
√
3
−
2
1 2
,
√
3
)
2
√ √ 2 , 22 ) 2
90◦ , π2
1
(2,
3
)
2
(
60◦ , π3
120◦ , 23π
−
(
√
3
2
, 12 )
30◦ , π6
150◦ , 56π
( 1, 0)
√ √ 2 , 22 ) 2
45◦ , π4
135◦ , 34π
, 12 )
√
180◦ , π 0◦ , 2π
210◦ , 76π (
√
3
− ,− 2
(
1
) 2
225◦ , 5π √
2
2
2
2
(
4
240◦ , 43π
300◦ , 53π
)
( 1
√
2
2
− ,−
(
315◦ , 7π
4
√
− ,−
330◦ , 116π
3
1
)
(2,
270◦ , 3π 2
−
√
(0, 1)
−
For any ordered pair on the unit circle (x, y ) : c o s θ = x and sin θ = y
Example
cos ( 76π )
=
√ 3
−2
sin ( 76π ) =
− 12
3
2
)
√
2
2
,
−
√
√
2
2
3
2
)
,
−
1 2
)
(1, 0)
Inverse Trig Functions Definition
Inverse Properties
θ = sin−1 (x) is equivalent to x = sin θ
These properties hold for x in the domain and θ in the range
θ = cos−1 (x) is equivalent to x = cos θ θ = tan−1 (x) is equivalent to x = tan θ
Domain and Range Function
θ = sin−1 (x) θ = cos−1 (x) θ = tan−1 (x)
Domain
sin(sin−1 (x)) = x
sin−1 (sin(θ)) = θ
cos(cos−1 (x)) = x
cos−1 (cos(θ)) = θ
tan(tan−1 (x)) = x
tan−1 (tan(θ)) = θ
Range
π
Other Notations
π
−1 ≤ x ≤ 1 − 2 ≤ θ ≤ 2 0 ≤ θ ≤ π −1 ≤ x ≤ 1 −∞ ≤ x ≤ ∞ − π2 < θ < π2
sin−1 (x) = arcsin(x) cos−1 (x) = arccos(x) tan−1 (x) = arctan(x)
Law of Sines, Cosines, and Tangents β a
c
γ
α b
Law of Sines
sin α a
=
sin β b
=
Law of Tangents
sin γ c
tan 12 (α β ) a b = a + b tan 12 (α + β )
−
−
Law of Cosines
a2 = b 2 + c2 b2 = a 2 + c2 c2 = a 2 + b2
− 2bc cos α − 2ac cos β − 2ab cos γ
tan 12 (β γ ) b c = b + c tan 12 (β + γ )
−
−
tan 12 (α γ ) a c = a + c tan 12 (α + γ )
−
−
Complex Numbers
√ −1
i =
√ −a = i√ a, a ≥ 0
i2 =
−1
i3 =
i4 = 1
−i
(a + bi)(a
− bi) = a √
(a + bi) + ( c + di) = a + c + ( b + d)i
|a + bi| =
(a + bi)
(a + bi) = a
− (c + di) = a − c + (b − d)i (a + bi)(c + di) = ac − bd + ( ad + bc)i
2
+ b2
a2 + b2 Complex Modulus
− bi Complex Conjugate (a + bi)(a + bi) = |a + bi| 2
DeMoivre’s Theorem Let z = r (cos θ + i sin θ), and let n be a positive integer. Then: z n = r n (cos nθ + i sin nθ). Example: Let z = 1
6
− i, find z .
Solution: First write z in polar form. r =
(1)2 + ( 1)2 =
−
2
− − √ − 1 1
θ = arg (z ) = tan−1
Polar Form: z =
√
π
=
2 cos
4
π
4
+ i sin
Applying DeMoivre’s Theorem gives : 6
4
√ · − · − − − 6
z =
π
−
2
cos 6
= 23 cos
= 8(0 + i(1)) = 8i
3π 2
π
4
+ i sin 6
+ i sin
3π 2
π
4
Finding the nth roots of a number using DeMoivre’s Theorem Example: Find all the complex fourth roots of 4. That is, find all the complex solutions of x4 = 4.
We are asked to find all complex fourth roots of 4. These are all the solutions (including the complex values) of the equation x4 = 4. For any positive integer n , a nonzero complex number z has exactly n distinct nth roots. More specifically, if z is written in the trigonometric form r(cos θ + i sin θ), the nth roots of z are given by the following formula.
( )r
∗
1 n
cos
θ 360 ◦ k + n n
+ i sin
θ 360 ◦ k + n n
,
f or k = 0, 1, 2,...,n
− 1.
Remember from the previous example we need to write 4 in trigonometric form by using: r =
(a)2 + (b)2
and
b . a
θ = arg (z ) = tan−1
So we have the complex number a + ib = 4 + i0. Therefore a = 4 and b = 0
So r =
(4)2 + (0)2 = 4 and 0 =0 θ = arg (z ) = tan−1 4 Finally our trigonometric form is 4 = 4(cos 0◦ + i sin0◦ )
Using the formula ( ) above with n = 4, we can find the fourth roots of 4(cos 0◦ + i sin0◦ )
∗
• For k = 0,
4
• For k = 1,
4
• For k = 2,
4
• For k = 3,
4
1 4
1 4
1 4
1 4
cos
cos cos cos
∗ ∗ ∗
0◦ 360◦ + 4 4 0◦ 360◦ + 4 4 ◦ 0 360◦ + 4 4 0◦ 360◦ + 4 4
∗0 1 2 3
Thus all of the complex roots of x4 = 4 are:
√ √ 2,
2i,
√ √ − 2, − 2i .
∗ ∗ ∗
0◦ 360◦ + i sin + 4 4 0◦ 360◦ + i sin + 4 4 ◦ 0 360◦ + i sin + 4 4 0◦ 360◦ + i sin + 4 4
∗0 1 2 3
=
√
=
√
2 (cos(0◦ ) + i sin(0◦ )) =
√
2 (cos(90◦ ) + i sin(90◦ )) =
√
2i
√ − 2 √ √ ◦ ◦ = 2 (cos(270 ) + i sin(270 )) = − 2i =
√
2
2 (cos(180◦ ) + i sin(180◦ )) =
Formulas for the Conic Sections
Circle
StandardForm : (x
2
− h)
+ (y
2
− k)
= r 2
Where (h, k) = center and r = radius
Ellipse
Standard Form for Horizontal Major Axis :
(x
2
− h) a2
+
(y
2
− k) b2
= 1
Standard Form for V ertical Major Axis :
(x
2
− h) b2
+
(y
2
− k) a2
= 1
Where (h, k)= center 2a=length of major axis 2b=length of minor axis
(0 < b < a ) Foci can be found by using c2 = a 2 Where c= foci length
2
−b
More Conic Sections Hyperbola
Standard Form for Horizontal Transverse Axis :
(x
2
2
− h) − (y − k) a b 2
2
= 1
Standard Form for V ertical Transverse Axis :
(y
2
2
− k) − (x − h) a b 2
2
= 1
Where (h, k)= center a=distance between center and either vertex
Foci can be found by using b2 = c 2
−a
Where c is the distance between center and either focus. ( b > 0 )
Parabola
Vertical axis: y = a (x
2
− h) + k Horizontal axis: x = a (y − k) + h 2
Where (h, k)= vertex a=scaling factor
2
f (x)
f (x) = sin(x)
1 √
3
2
√
2
2 1 2
x
0
π
π
π
6
4
3
π
2
2π 3
3π 4
5π 6
π
7π 6
5π 4
4π 3
3π 2
5π 3
7π 4
11π 6
2π
4π 3
3π 2
5π 3
7π 4
11π 6
2π
1
−
2
√
2
− √ − 2
3
2
-1
√ 5 2 π Example : sin = − 4
2
f (x)
f (x) = cos(x)
1 √
3 2
√
2
2 1 2
x
0
π
π
π
6
4
3
π
2
2π 3
3π 4
5π 6
π
7π 6
5π 4
1
−
2
√
2
− √ − 2
3
2
-1
√ 7 3 π Example : cos = − 6
2
f (x)
π
−
2
π
2
f (x) = tan x √
3
1 √
3
3
−π
5π 6
3π 4
− − −
2π 3
π
π
− − − 3
4
0
π
6
√
3
−
3
−1 √ − 3
x π
π
π
6
4
3
2π 3
3π 4
5π 6
π
