Download Common Trigonometric Angle Measurements

Common Trigonometric Angle Measurements
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
Hypotenuse =
Angle in Degrees
Angle in Radians
0
0
SIN
0
COS
1
TAN
0
CSC
ND
SEC
1
COT
ND
30
π
6
1
2
√3
2
√3
3
2
45
π
4
√2
2
√2
2
1
60
π
3
√3
2
1
2
√3
√2
2√3
3
√3
√2
2√3
3
2
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1
√3
3
90
π
2
1
0
ND
1
ND
0
120
2π
3
√3
2
-1
2
-√3
135
3π
4
√2
2
-√2
2
-1
2√3
3
-2
√2
-√3
3
-√2
-1
1
150
5π
6
1
2
-√3
2
-√3
3
2
180
π
-2√3
3
-√3
-1
0
-1
0
ND
ND
210
7π
6
-1
2
-√3
2
√3
3
-2
225
5π
4
-√2
2
-√2
2
1
240
4π
3
-√3
2
-1
2
√3
270
3π
2
-1
-√2
-1
-2√3
3
√3
-√2
-2√3
3
-2
1
√3
3
0
ND
ND
0
300
5π
3
-√3
2
1
2
-√3
315
7π
4
-√2
2
√2
2
-1
-2√3
3
2
-√2
-√3
3
-1
√2
330
11π
6
-1
2
√3
2
-√3
3
-2
360
2π
2√3
3
-√3
1
0
1
0
ND
ND
Common Trigonometric Angle Measurements
Reviewed July 2008
Angles
sin and csc
tan and cot
cos and sec
Q1
+
+
+
Q2
+
-
Q3
+
-
Reference Angles θ΄
Q4
+
If…..
0º θ ≤ 90
then….. θ΄ = θ
90º < θ ≤ 180 º
θ΄ = 180º - θ
180º < θ ≤ 270º
θ΄ = θ - 180º
270º < θ ≤ 360º
θ΄ = 360º - θ
To convert from Radians to
Degrees……
Multiply by
When using the unit circle:
To convert from Degrees to
Radians…….
Multiply by
Positive angles move
counter clockwise.
Start at 0° and then….
Negative angles move
clockwise.
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Common Trigonometric Angle Measurements
Fundamental Identities
tan θ =
cot θ =
sin2 θ + cos2 θ = 1
csc θ =
1 + tan2 θ = sec2 θ
sec θ =
cot θ =
1 + cot2 θ = csc2 θ
Sum and Difference Identities
cos (α – β) = cos α cos β + sin α sin β
tan (α – β) =
cos (α + β) = cos α cos β – sin α sin β
sin (α – β) = sin α cos β – cos α sin β
tan (α + β) =
sin (α + β) = sin α cos β + cos α sin β
Double-Angle Identities
sin 2α = 2 sin α cos α
cos 2α = cos2α – sin2α = 1 – 2sin2α = 2cos2α – 1
tan 2α =
Half-Angle Identities
sin
= ±
cos
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= ±
tan
=
3
=
=
Common Trigonometric Angle Measurements
Even and Odd Identities
sin (-α) = - sin α
cos (-α) = cos α
tan (-α) = - tan α
csc (-α) = - csc α
sec (-α) = sec α
cot (-α) = - cot α
Power-Reducing Identities
sin2 α =
cos2 α =
tan2 α =
Product-to-Sum Identities
Sum-to-Product Identities
For sums of the form a sin x + b cos x ……..
Where
,
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,
and
4
Common Trigonometric Angle Measurements
Linear Speed (υ)
Arc Length Formula
s=rθ
or
θ = radians
υ=
s = length of the arc
r = radius of the circle
θ = non-negative radian measure of the central angle
s = distance traveled
t = time
Angular Speed (ω)
ω=
θ = measure of angle in radians
t = time
Converting from Degrees to Minutes to Seconds
1° (degree) = 60´ (minutes)
Linear Speed – Angular Speed Relationship
1´ (minute) = 60˝ (seconds)
υ = rω
1° (degree) = 3,600˝ (seconds)
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υ = linear speed
r = radius
ω = angular speed
Common Trigonometric Angle Measurements
Law of Sines: used to solve triangles when two angles and a side are given or when two sides and an opposite angle are given.
Law of Cosines: used to solve triangles when two sides and the included angle or three sides of the triangle are given.
Area (K) using Heron’s Formula
Area (K)
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Common Trigonometric Angle Measurements