Trigonometry Formula Sheet PDF

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Trigonometric Formula Sheet All formulas below must be memorized unless it is given to you on your formula sheet. Right Triangle Trigonometry

Unit Circle Trigonometry

Degrees to Radians 𝑥° 𝜋 ) = 𝑟𝑎𝑑𝑖𝑎𝑛 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 ( )×( 180° 1

Radians to Degrees

(

sin 𝜃 =

𝑜𝑝𝑝

ℎ𝑦𝑝

Definition of Inverse Functions

csc 𝜃 = 𝑜𝑝𝑝

ℎ𝑦𝑝

𝜃 = sin−1(𝑥) is equal to 𝑥 = sin 𝜃

ℎ𝑦𝑝

𝑎𝑑𝑗

sec 𝜃 = 𝑎𝑑𝑗

cos 𝜃 = ℎ𝑦𝑝 𝑜𝑝𝑝

𝜃 = cos−1(𝑥) is equal to 𝑥 = cos 𝜃

𝑎𝑑𝑗 𝑜𝑝𝑝

cot 𝜃 =

tan 𝜃 = 𝑎𝑑𝑗

𝑦

sin 𝜃 = 𝑟 =

Tangent & Cotangent Identities sin 𝜃

tan 𝜃 = cos 𝜃

cot 𝜃 =

𝑥

cos 𝜃 = = 𝑟

cos 𝜃 sin 𝜃

𝑦

tan 𝜃 = 𝑥

Reciprocal Identities

sin 𝜃 =

cos 𝜃 =

tan 𝜃 =

1

csc 𝜃 1

sec 𝜃 1 cot 𝜃

csc 𝜃 =

sec 𝜃 =

cot 𝜃 =

1 sin 𝜃

𝑦 1

csc 𝜃 =

𝑥 1

sec 𝜃

𝜋

𝜋

csc ( − 𝜃) = sec 𝜃 2 𝜋

tan (2 − 𝜃) = cot 𝜃

𝑟 =𝑥

Pythagorean Identities

= 𝑥 𝑦

Other Notations

sin−1(𝑥) = arcsin(𝑥)

1 𝑥

cos−1(𝑥) = arccos(𝑥)

tan−1(𝑥) = arctan(𝑥)

sin(sin−1(𝑥)) = 𝑥

tan2 𝜃 + 1 = sec2 𝜃

1 tan 𝜃

1

=𝑦

𝜃 = tan−1(𝑥) is equal to 𝑥 = tan 𝜃

Inverse Properties

sin2 𝜃 + cos2 𝜃 = 1

1 cos 𝜃

𝑟 𝑦

cot 𝜃 =

cos(cos−1 (𝑥)) = 𝑥

1 + cot 2 𝜃 = csc2 𝜃

Cofunction Formulas sin ( − 𝜃) = cos 𝜃 2

𝑥𝜋 180° ) = 𝑑𝑒𝑔𝑟𝑒𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 )×( 𝜋 1

tan(tan−1(𝑥)) = 𝑥

sin−1(sin(𝑥)) = 𝑥

cos−1(cos(𝑥)) = 𝑥 tan−1(tan(𝑥)) = 𝑥

Domain and Range of Inverse Functions

𝜋

cos ( 2 − 𝜃) = sin 𝜃 𝜋

sec ( 2 − 𝜃) = csc 𝜃 𝜋

cot ( 2 − 𝜃) = tan 𝜃

Fu Functi ncti nctio on

𝜃=

𝜃=

sin−1(𝑥)

cos−1(𝑥)

𝜃 = tan−1(𝑥)

Even and Odd Formulas

Do Domai mai main n

−1 ≤ 𝑥 ≤ 1 −1 ≤ 𝑥 ≤ 1

−∞ ≤ 𝑥 ≤ ∞

Ran Range ge 𝜋 𝜋 − ≤𝜃≤ 2 2 0≤𝜃≤𝜋 𝜋 𝜋 − ≤𝜃≤ 2 2

Qua Quad dran rants ts I & IV I & II I & IV

Sum and Difference Formulas

sin(−𝜃) = − sin 𝜃

csc(−𝜃) = − csc 𝜃

sin(𝛼 ± 𝛽) = sin 𝛼 cos 𝛽 ± cos 𝛼 sin 𝛽

tan(−𝜃) = − tan 𝜃

cot(−𝜃) = − cot 𝜃

cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽

cos(−𝜃) = cos 𝜃

sec(−𝜃) = sec 𝜃

On the back you will find a completed unit circle that you will have to be able to do on your own.

“sine, cosine, cosine, sine, same sign”

“cosine, cosine, sine, sine, sw switc itc itch h sign” tan(𝛼 ± 𝛽) =

tan 𝛼 ± tan 𝛽 1 ∓ tan 𝛼 tan 𝛽

“tangent is a mess, and very self-centered”

The Unit Circle

Finding Reference Angles

You must be able to complete the unit circle on your own for all four quadrants.

Step 1: Determine the quadrant your angle is located in. Step 2: Subtract using the closest 𝑥 – 𝑎𝑥𝑖𝑠 angle measure.

Key Information from your Unit Circle Graph (0, 1)

(-1, 0)

S

A

T

C

1, 0)

(0, -1)

Product to Sum Formulas

Table of Unit Circle Values

*Given on Formula Sheet*

Quadrant I

sin 𝛼 sin 𝛽 =

cos 𝛼 cos 𝛽 =

sin 𝛼 cos 𝛽 =

1 [cos(𝛼 − 𝛽) − cos(𝛼 + 𝛽)] 2 1

2

1

2

[cos(𝛼 − 𝛽) + cos(𝛼 + 𝛽 )]

[sin(𝛼 + 𝛽) + sin(𝛼 − 𝛽)]

1 cos 𝛼 sin 𝛽 = [sin(𝛼 + 𝛽) − sin(𝛼 − 𝛽)] 2

sin 𝜃

cos 𝜃

tan 𝜃

𝜋 (30°) 6 1 2 √3 2 √3 3

𝜋 (45°) 4 √2 2 √2 2 1

𝜋 (60°) 3 √3 2 1 2 √3

Double Angle Formulas

Half Angle Formulas

Sum to Product Formulas

*Given on Formula Sheet*

*Given on Formula Sheet*

*Given on Formula Sheet*

cos 2𝜃 = cos2 𝜃 − sin2 𝜃

1 − cos 2𝜃 sin 𝜃 = ±√ 2

sin 2𝜃 = 2 sin 𝜃 cos 𝜃 =

2 cos2 𝜃

−1

= 1 − 2 sin2 𝜃

tan 2𝜃 =

2 tan 𝜃 1 − tan2 𝜃

1 + cos 2𝜃 cos 𝜃 = ±√ 2

1 − cos 2𝜃 tan 𝜃 = ±√ 1 + cos 2𝜃

sin 𝛼 + sin 𝛽 = 2 sin (

𝛼−𝛽 𝛼+𝛽 ) cos ( ) 2 2

sin 𝛼 − sin 𝛽 = 2 cos (

cos 𝛼 + cos 𝛽 = 2 cos (

𝛼−𝛽 𝛼+𝛽 ) ) sin ( 2 2

𝛼+𝛽 𝛼−𝛽 ) cos ( ) 2 2

cos 𝛼 − cos 𝛽 = −2 sin (

𝛼−𝛽 𝛼+𝛽 ) sin ( ) 2 2...