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Trigonometry Formula Sheet: Definitions, Identities, and Applications, Exams of Trigonometry

Trigonometric Formula Sheet Definition of the Trig Functions

Typology: Exams

2022/2023

Available from 03/26/2024

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1
2
2
2
(x, y)
1y
θ
x
Trigonometric Formula Sheet
Definition of the Trig Functions
Right Triangle Definition
Assume that:
Unit Circle Definition
Assume θ can be any angle.
0
<
θ
<
π
or 0
<
θ
<
90
y
hypotenuse
θ
adjacent
opposite x
sin θ
=
opp
hyp csc θ
=
hyp
opp
sin θ
=
y
1
csc θ
=
1
y
cos θ =
adj
hyp
sec θ =
hyp
adj
cos θ =
x
1
sec θ =
1
x
tan θ
=
opp
adj
cot θ
=
adj
opp
tan θ
=
y
x
cot θ
=
x
y
sin θ,
θ (−∞, )
cos θ, θ (−∞, )
Domains of the Trig Functions
csc θ,
θ /= nπ, where n Z
sec θ,
θ n
+
1
π, where n Z
tan θ,
θ
n
+
1
π, where n Z
cot θ,
θ
nπ, where n Z
Ranges of the Trig Functions
1 sin θ 1
1 cos θ 1
−∞ ≤ tan θ ≤ ∞
csc θ 1 and csc θ
1 sec θ 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.
2π
sin(ωθ) T
=
ω
2π
cos(ωθ) T
=
ω
π
2π
csc(ωθ) T
=
ω
2π
sec(ωθ) T
=
ω
π
tan(ωθ) T
=
ωcot(ωθ) T
=
ω
pf3
pf4
pf5
pf8
pf9
pfa

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Download Trigonometry Formula Sheet: Definitions, Identities, and Applications and more Exams Trigonometry in PDF only on Docsity!

2

(x, y)

y

θ

x

Trigonometric Formula Sheet

Definition of the Trig Functions

Right Triangle Definition

Assume that:

Unit Circle Definition

Assume θ can be any angle.

0 < θ <

π

or 0

< θ < 90

y

hypotenuse

θ

adjacent

opposite

x

sin θ =

opp

hyp

csc θ =

hyp

opp

sin θ =

y

csc θ =

y

cos θ =

adj

hyp

sec θ =

hyp

adj

cos θ =

x

sec θ =

x

tan θ =

opp

adj

cot θ =

adj

opp

tan θ =

y

x

cot θ =

x

y

sin θ,θ ∈ (−∞ , ∞)

cos θ,θ ∈ (−∞ , ∞)

Domains of the Trig Functions

csc θ,θ /= nπ, where n ∈ Z

sec θ,θ n +

π, where n ∈ Z

tan θ,θ

n +

π, where n ∈ Z

cot θ,θ nπ, where n ∈ Z

Ranges of the Trig Functions

− 1 ≤ sin θ ≤ 1

− 1 ≤ cos θ ≤ 1

−∞ ≤ tan θ ≤ ∞

csc θ ≥ 1 and csc θ

− 1 sec θ ≥ 1 and sec θ

−∞ ≤ 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.

2 π sin( ωθ ) ⇒ T =

ω

2 π cos( ωθ ) ⇒ T =

ω

π

2 π csc( ωθ ) ⇒ T =

ω

2 π sec( ωθ ) ⇒ T =

ω

π

tan( ωθ ) ⇒ T =

ω

cot( ωθ ) ⇒ T =

ω

r

Identities and Formulas

Tangent and Cotangent Identities Half Angle Formulas

sin θ

tan θ =

cos θ

cos θ

cot θ =

sin θ

sin θ = ±

1 − cos(2 θ )

Reciprocal Identities

cos θ = ±

r

1 + cos(2 θ )

s

1 − cos(2 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 and Odd Formulas

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(− θ ) = − sin θ

cos(− θ ) = cos θ

tan(− θ ) = − tan

θ

Periodic Formulas

If n is an integer

sin( θ + 2 πn ) = sin θ

cos( θ + 2 πn ) = cos θ

tan( θ + πn ) = tan θ

csc(− θ ) = − csc

θ sec(− θ ) = sec θ

cot(− θ ) = − cot

θ

csc( θ + 2 πn ) = csc θ

sec( θ + 2 πn ) = sec θ

cot( θ + πn ) = cot θ

sin α sin β =

[cos( αβ ) − cos( α + β )]

cos α cos β =

[cos( αβ ) + cos( α + β )]

sin α cos β =

[sin( α + β ) + sin( αβ )]

cos α sin β =

[sin( α + β ) − sin( αβ )]

Sum to Product Formulas

sin α + sin β = 2 sin

α + β

cos

α

β

sin(2 θ ) = 2 sin θ cos θ

cos(2 θ ) = cos

2

θ − sin

2

θ

sin α − sin β = 2 cos

α + β

sin

αβ

cos α + cos β = 2 cos

α + β

cos

αβ

= 2 cos

2

θ

= 1 − 2 sin

2

θ

2 tan θ

cos α − cos β = − 2

sin

α + β

sin

αβ

tan(2 θ ) =

1 − tan

2

θ

Degrees to Radians Formulas

If x is an angle in degrees and t is an angle in

radians then:

Cofunction Formulas

sin

π

θ = cos θ

csc

π

θ = sec θ

cos

π

θ = sin θ

sec

π

θ = csc θ

π t

πx

t =

t

and x =

tan

π

θ = cot

θ

tan θ =

1 + cos(2 θ )

Double Angle Formulas

x

π

sin θ =

csc θ

csc θ =

sin θ

cos θ =

sec θ

sec θ =

cos θ

tan θ =

cot θ

cot θ =

tan θ

13

90 ◦, π

2

22

13

√ 2 √ 2 √ 2 √ 2

120 ◦, 2 π

3

60 ◦, π

3

3 1

135 ◦, 3 π 45 ◦, π

22

4

4

22

150 ◦, 5 π

6

30 ◦, π

6

180 ◦, π

0 ◦, 2π

210 ◦, 7 π

6

330 ◦, 11 π

6

√ 3

1

√ 3

2 2

225 ◦, 5 π

4

315 ◦, 7 π

4

1

2 2

240 ◦, 4 π

3

300 ◦, 5 π

3

√ 2 √ 2

√ 2 √ 2

1

√ 3 √ 3

22

270 ◦, 3 π

1

2

2

2

Unit Circle

For any ordered pair on the unit circle ( x, y ) : cos θ = x and sin θ = y

Example

cos (

7 π

) = −

sin (

7 π

) = −

2

2

2

2

2

β

a c

γ

α

Inverse Trig Functions

Definition

θ = sin

1

( x ) is equivalent to x = sin θ

Inverse Properties

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

sin(sin

1

( x )) = x

cos(cos

1

( x )) = x

tan(tan

1

( x )) =

x

sin

1

(sin( θ )) = θ

cos

1

(cos( θ )) = θ

tan

1

(tan( θ )) = θ

Function

θ = sin

1

( x )

θ = cos

1

( x )

θ = tan

1

( x )

Domain

− 1 ≤ x ≤ 1

− 1 ≤ x ≤ 1

−∞ ≤ x ≤ ∞

Range

π π

θ

0 ≤ θ

π π π

< θ <

Other Notations

sin

1

( x ) = arcsin( x )

cos

1

( x ) = arccos( x )

tan

1

( x ) = arctan( x )

Law of Sines, Cosines, and Tangents

b

Law of Sines Law of Tangents

sin α sin β sin

γ

ab

tan

1

( αβ )

a b

c

Law of Cosines

a +

b

bc

tan

1

( α + β )

tan

1

( βγ )

a

2

= b

2

  • c

2

− 2 bc cos

α

b + c

2

tan

1

( β + γ )

b

2

= a

2

  • c

2

− 2 ac cos

β

ac

tan

1

( αγ )

c

2

= a

2

  • b

2

− 2 ab cos γ

a + c

a = i

a, a ≥ 0

Complex Numbers

i =

− 1 i

2

= − 1 i

3

= − i i

4

( a + bi )( abi ) = a

2

b

2

( a + bi ) + ( c + di ) = a + c + ( b +

d ) i ( a + bi ) − ( c + di ) = ac + ( b

d ) i ( a + bi )( c + di ) = acbd + ( ad +

bc ) i

| a + bi | =

a

2

  • b

2

Complex Modulus

( a + bi ) = abi 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:

Example: Let z = 1 − i , find z

6

z

n

= r

n

(cos + i sin ).

Solution: First write z in polar form.

r =

2

2

θ = arg ( z ) = tan

1

π

Polar Form: z =

2 cos −

π

  • i sin −

π

Applying DeMoivre’s Theorem gives :

z

6

6

cos 6 · −

π

  • i sin 6 · −

π

3

cos −

3 π

  • i sin −

3 π

= 8(0 + i (1))

= 8 i

n

1

θ = arg ( z ) = tan

4

4

4

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

x

4

We are asked to find all complex fourth roots of 4.

These are all the solutions (including the complex values) of the equation x

4

For any positive integer n , a nonzero complex number z has exactly n distinct n th roots.

More specifically, if z is written in the trigonometric form r (cos θ + i sin θ ), the n th roots of

z are given by the following formula.

(∗) r

1

cos

k

+ i sin

k

, for k = 0 , 1 , 2 , ..., n − 1.

n n n n

Remember from the previous example we need to write 4 in trigonometric form by using:

r = ( a )

2

  • ( b )

2

and θ = arg ( z ) = tan

1

b

a

So we have the complex number a + ib = 4 + i 0.

Therefore a = 4 and b = 0

So r =

2

2

= 4 and

Finally our trigonometric form is 4 = 4(cos 0

  • i sin 0

Using the formula (∗) above with n = 4, we can find the fourth roots of 4(cos 0

  • i sin 0

For k = 0 , 4 cos + + i sin + =

2 (cos(

) + i sin(

For k = 1 , 4

1

cos

  • i sin

2 (cos(

) + i sin(

2 i

For k = 2 , 4

1

cos

  • i sin

2 (cos(

) + i sin(

For k = 3 , 4

1

cos

  • i sin

2 (cos(

) + i sin(

2 i

Thus all of the complex roots of x

4

= 4 are:

2i ,

2i.

More Conic Sections

Hyperbola

Standard Form for Horizontal Transverse Axis :

( x h )

2

a

2

( y k )

2

b

2

Standard Form for V ertical Transverse Axis :

( y k )

2

a

2

( x h )

2

b

2

Where ( h , k )= center

a =distance between center and either vertex

Foci can be found by using b

2

= c

2

− a

2

Where c is the distance between

center and either focus. ( b > 0 )

Parabola

Vertical axis: y = a ( x − h )

2

+ k

Horizontal axis: x = a ( y − k )

2

+ h

Where ( h , k )= vertex

a =scaling factor

f (x) = sin(x)

1

√ 3

2

√ 2

2

1

2

0 πππ

643

π

2

2π3π5π

3 4 6

π

7π5π4π

64 3

2

5π7π 11π

3 4 6

1

2

— 2

√ 2

— 2

-

√ 3

Example : sin



4

= −

2

2

f (x) = cos(x)

1

√ 3

2

√ 2

2

1

2

0 πππ

643

π

2

2π3π5π

346

π

7π5π4π

643

2

5π7π 11π

346

1

2

— 2

√ 2

— 2

-

√ 3

Example : cos



6

= −

3

2

f ( x )

x

f ( x )

x