TRIGONOMETRY

ANGLE MEASUREMENT

π radius = 180°

1° =𝜋

180rad 1 rad =

180°

𝜋

s = r θ

(θ in radians)

RIGHT ANGLE TRIGONOMETRY

sin 𝜃 = opp

ℎyp csc 𝜃 =

ℎyp

opp

cos 𝜃 = 𝑎𝑑𝑗

hyp sec 𝜃 =

ℎyp

𝑎𝑑𝑗

tan 𝜃 = 𝑜𝑝𝑝

adj csc 𝜃 =

𝑎𝑑𝑗

opp

TRIGONOMETRIC FUNCTIONS

sin 𝜃 = y

𝑟 csc 𝜃 =

r

𝑦

cos 𝜃 = x

𝑟 sec 𝜃 =

r

𝑥

tan 𝜃 = y

𝑥 cot 𝜃 =

x

𝑦

GRAPHS OF THE TRIGONOMETRIC FUNCTIONS

TRIGONOMETRIC FUNCTIONS OF IMPORTANT

ANGLES

θ radius sin θ cos θ tan θ

0° 0 0 1 0

30° π/6 1/2 √3/2 √3/3

45° π/4 √2/2 √2/2 1

60° π/3 √3/2 1/2 √3

90° π/2 1 0 --

FUNDMENTAL IDENTITIES

csc 𝜃 = 1

sin 𝜃 sec 𝜃 =

1

cos 𝜃

tan 𝜃 = sin 𝜃

cos 𝜃 cot 𝜃 =

cos 𝜃

sin 𝜃

cot 𝜃 = 1

tan 𝜃 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2𝜃 = 1

1 + 𝑡𝑎𝑛2 𝜃 = sec2𝜃 1 + 𝑐𝑜𝑡2 𝜃 = csc2𝜃

sin(−𝜃) = − sin 𝜃 cos(−𝜃) = cos θ

tan(−𝜃) = − tan𝜃 sin(𝜋

2− 𝜃) = cosθ

cos(𝜋

2− 𝜃) = sinθ tan(

𝜋

2− 𝜃) = cotθ

ADDITION AND SUBTRACTION FORMULAS

sin(𝑥 + 𝑦) = 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 + 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑦

sin(𝑥 − 𝑦) = 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 − 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑦

cos(𝑥 + 𝑦) = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦

cos(𝑥 − 𝑦) = 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 + 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦

DOUBLE-ANGLE FORMULAS

sin 2𝑥 = 2 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥

cos 2𝑥 = 𝑐𝑜𝑠2𝑥 −𝑠𝑖𝑛𝑥2

= 2𝑐𝑜𝑠𝑥2 − 1 = 1 − 2𝑠𝑖𝑛2𝑥

tan 2𝑥 =2 tan 𝑥

1 − 𝑡𝑎𝑛2 𝑥

HALF-ANGLE FORMULAS

𝑠𝑖𝑛2 𝑥 =1−cos 2𝑥

2 𝑐𝑜𝑠2 𝑥 =

1+cos 2𝑥

2

s

r θ

4

4

4

4

θ r

θ

θ

hyp opp

adj

θ

4

4

4

4

θ

(x,y)

4444

θ

-1

1

-1

1

-1

1

-1

1

x

y

x

y

x

y

x

y

x

y

x

y

2π π

π 2π

π 2π π 2π

π 2π

π 2π