trigonometrycguhueililin.myds.me/download/proofs.pdf · 2019-12-27 · trigonometry angle...
TRANSCRIPT
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