Common Trigonometric Angle Measurements
𝑜𝑝𝑝
sin θ = ℎ𝑦𝑝
𝑎𝑑𝑗
cos θ = ℎ𝑦𝑝
𝑜𝑝𝑝
tan θ = 𝑎𝑑𝑗
ℎ𝑦𝑝
ℎ𝑦𝑝
csc θ = 𝑜𝑝𝑝
𝑎𝑑𝑗
sec θ = 𝑎𝑑𝑗
cot θ = 𝑜𝑝𝑝
Hypotenuse = √𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 2 + 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 2
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 March 2015
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……
180°
Multiply by π 𝒓𝒂𝒅𝒊𝒂𝒏𝒔
When using the unit circle:
To convert from Degrees to
Radians…….
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
Multiply by 180°
Positive angles move
counter clockwise.
Start at 0° and then….
Negative angles move
clockwise.
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Common Trigonometric Angle Measurements
Fundamental Identities
sin 𝜃
tan θ = cos 𝜃
cot θ =
sin2 θ + cos2 θ = 1
cos 𝜃
1
1
csc θ = sin 𝜃
sin 𝜃
1 + tan2 θ = sec2 θ
1
sec θ = cos 𝜃
cot θ = tan 𝜃
1 + cot2 θ = csc2 θ
Sum and Difference Identities
tan 𝛼−tan 𝛽
cos (α – β) = cos α cos β + sin α sin β
tan (α – β) = 1+tan 𝛼 tan 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) = 1−tan 𝛼 tan 𝛽
tan 𝛼+tan 𝛽
sin (α + β) = sin α cos β + cos α sin β
Double-Angle Identities
2 tan 𝜃
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ – sin2 θ = 1 – 2sin2 θ = 2cos2 θ – 1
tan 2θ = 1−𝑡𝑎𝑛2 𝜃
Half-Angle Identities
𝛼
1−cos 𝛼
sin 2 = ± √
2
𝛼
1+cos 𝛼
cos 2 = ± √
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2
𝛼
tan 2 =
3
sin 𝛼
1+cos 𝛼
=
1−cos 𝛼
sin 𝛼
𝛼
= ±√1−cos
1+cos 𝛼
Common Trigonometric Angle Measurements
Even and Odd Identities
sin (-θ) = - sin θ
cos (-θ) = cos θ
tan (-θ) = - tan θ
csc (-θ) = - csc θ
sec (-θ) = sec θ
cot (-θ) = - cot θ
Power-Reducing Identities
sin2 θ =
1−cos 2𝜃
cos2 θ =
2
1+cos 2𝜃
1−cos 2𝜃
tan2 θ= 1+cos 2𝜃
2
Product-to-Sum Identities
sin α cos β = 12[sin(α + β) + sin(α − β)]
cos 𝛼 cos 𝛽 = 12 [cos(𝛼 + 𝛽) + cos(𝛼 − 𝛽)]
cos 𝛼 sin 𝛽 = 12[sin(𝛼 + 𝛽) − sin(𝛼 − 𝛽)]
sin 𝛼 sin 𝛽 = 12 [cos(𝛼 − 𝛽) − cos(𝛼 + 𝛽)]
Sum-to-Product Identities
sin 𝛼 + sin 𝛽 = 2 𝑠𝑖𝑛
𝛼+𝛽
cos 𝛼 + cos 𝛽 = 2 𝑐𝑜𝑠
2
𝑐𝑜𝑠
𝛼+𝛽
2
𝛼−𝛽
𝑐𝑜𝑠
sin 𝛼 − sin 𝛽 = 2 𝑐𝑜𝑠
2
𝛼−𝛽
𝛼+𝛽
2
cos 𝛼 − cos 𝛽 = −2𝑠𝑖𝑛
2
𝑠𝑖𝑛
𝛼+𝛽
2
𝛼−𝛽
2
𝑠𝑖𝑛
𝛼−𝛽
2
For sums of the form a sin x + b cos x ……..
𝑎 sin 𝑥 + 𝑏 cos 𝑥 = 𝑘 sin(𝑥 + 𝛼)
𝑏
Where 𝑘 = √𝑎2 + 𝑏 2 , sin 𝛼 = √𝑎2
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+𝑏 2
,
and cos 𝛼 =
4
𝑎
√𝑎2 +𝑏 2
Common Trigonometric Angle Measurements
Linear Speed (υ)
Arc Length Formula
s=rθ
or
𝑠
𝑠
θ = 𝑟 radians
υ=𝑡
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
θ = measure of angle in radians
t = time
𝑡
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.
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
Law of Cosines: used to solve triangles when two sides and the included angle or three sides of the triangle are given.
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
Area (K) using Heron’s Formula
Area (K)
𝐾=
1
𝑏 2 sin 𝐶 sin 𝐴
𝑏𝑐 sin 𝐴 =
2
2 sin 𝐵
1
𝐾 = 𝑎𝑏 sin 𝐶
2
1
𝐾 = 𝑎𝑐 sin 𝐵
2
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𝐾 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
1
𝑠 = 2(𝑎+𝑏+𝑐)
6
Common Trigonometric Angle Measurements