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Math 150 T17-Trig “Cheat Sheet”
Page 1
Pythagorean Identities
MATH 150 – TOPIC 17
TRIG “CHEAT SHEET”
sin2 x + cos2 x = 1, 1 + tan2 x = sec2 x,
1 + cot2 x = csc2 x
Definition of the Six Trigonometric Functions
Right triangle definitions, where 0 < θ < π/2.
Reduction Formulas
sin(−x) = − sin x
csc(−x) = − csc x
sec(−x) = sec x
Hypotenuse
Opposite
θ
cos(−x) = cos x
tan(−x) = − tan x
cot(−x) = − cot x
Adjacent
sin θ =
opposite
hypotenuse
csc θ =
hypotenuse
opposite
cos θ =
adjacent
hypotenuse
sec θ =
hypotenuse
adjacent
cos(u ± v) = cos u cos v ∓ sin u sin v
tan θ =
opposite
adjacent
cot θ =
adjacent
opposite
tan(u ± v) =
Unit circle definitions, where θ is any angle.
y
1
(x, y)
•
y
1=
p
1
x
•
x2
θ
1
+
y2
x
y
sin θ = = y
1
1
csc θ =
y
x
cos θ = = 1
1
y
tan θ =
x
1
sec θ =
x
x
cot θ =
y
Sum and Difference Formulas
sin(u ± v) = sin u cos v ± cos u sin v
tan u ± tan v
1 ∓ tan u tan v
Cofunction Identities
π
sin
− x = cos x
2
π
csc
− x = sec x
2
π
− x = csc x
sec
2
cos
tan
cot
π
2
π
2
π
2
− x = sin x
− x = cot x
− x = tan x
Double Angle Formulas
Thus, x = cos θ, y = sin θ
tan 2u =
Reciprocal Identities
1
csc x
1
csc x =
sin x
sin x =
1
cos x
1
cos x =
sec x
sec x =
1
cot x
1
cot x =
tan x
tan x =
Tangent and Cotangent Identities
tan x =
sin x
cos x
sin 2u = 2 sin u cos u
cos 2u = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u
cot x =
cos x
sin x
2 tan u
1 − tan2 u
Half Angle Formulas
1 − cos 2u
2
1 + cos 2u
cos2 u =
2
1 − cos 2u
tan2 u =
1 + cos 2u
sin2 u =
Continued on next page
Math 150 T17-Trig “Cheat Sheet”
Page 2
Product-to-Sum Formulas
1
sin u sin v = [cos(u − v) − cos(u + v)]
2
1
cos u cos v = [cos(u − v) + cos(u + v)]
2
1
sin u cos v = [sin(u + v) + sin(u − v)]
2
1
cos u sin v = [sin(u + v) − sin(u − v)]
2
Sum-to-Product Formulas
u+v
u−v
sin u + sin v = 2 sin
cos
2
2
u+v
u−v
sin u − sin v = 2 cos
sin
2
2
u−v
u+v
cos
cos u + cos v = 2 cos
2
2
u+v
u−v
cos u − cos v = −2 sin
sin
2
2
y
(− 12 ,
√
(−
√
(−
√
2
2
,
)
2
2
•
3 1
, )
2 2 • 5π
6
3π
4
√
(0, 1)
3
)
2
•
2π
3
120
◦
60
135◦
150◦
√
• 4
2
, − 22 )
2
3
)
2
π
4
◦
45◦
30◦
•
√
2
2
,
)
2
2
(
•
π
6
4π
3
•
√
1
3
(− 2 , − 2 )
270◦
3π
2
•
300
(0, −1)
5π
3
7π
4 •
√
•
x = cos θ
y = sin θ
√
3 1
, )
2 2
where θ is the indicated
angle.
330◦
11π
315◦
6 •
◦
For any pair (x, y),
√
(
0◦
0
◦ 2π
360
210◦
7π
√
225◦
• 6
3
1
(− 2 , − 2 )
240◦
5π
(−
90◦
√
•
π
3
180◦
(−1, 0) • π
√
( 12 ,
•
π
2
(
(
•
(1, 0)
√
3
, − 12 )
2
√
2
2
,
−
)
2
2
√
( 12 , − 23 )
x
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