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PreCalculus Exam Formula Sheets, Cheat Sheet of Calculus

Coastal Bend College Calculus

Final exam review and cheat sheet on precalculus

Typology: Cheat Sheet

2020/2021

Uploaded on 04/23/2021

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CBC MATHEMATICS

MATH 2412-PreCalculus

Exam Formula Sheets

CBC Mathematics 2019Fall

 System of Equations and Matrices

 3 Matrix Row Operations:

• Switch any two rows.

• Multiply any row by a nonzero constant.

• Add any constant-multiple row to another

 Even and Odd functions

• Even function: 𝑓(−𝑥)= 𝑓(𝑥) Odd function: 𝑓(−𝑥)= −𝑓(𝑥)

 Graph Symmetry

• 𝑥-axis symmetry: if (𝑥,𝑦) is on the graph, then (𝑥,−𝑦) is also on the graph

• 𝑦-axis symmetry: if (𝑥,𝑦) is on the graph, then (−𝑥,𝑦) is also on the graph

• origin symmetry: if (𝑥,𝑦)f is on the graph, then (−𝑥,−𝑦) is also on the graph

 Function Transformations

 Stretch and Compress

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

 Reflections

• 𝑦 = −𝑓(𝑥) reflect 𝑓(𝑥) about 𝑥-axis

• 𝑦 = 𝑓(−𝑥) reflect 𝑓(𝑥) about 𝑦-axis

 Stretch and Compress

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

: compress 𝑓(𝑥) if 0 < 𝑎 <1

• 𝑦 = 𝑓(𝑎𝑥), 𝑎 > 0 horizontal: stretch 𝑓(𝑥) if 0 < 𝑎 <1

: compress 𝑓(𝑥) if 𝑎 > 1

 Shifts

• 𝑦 = 𝑓(𝑥) +𝑘, 𝑘 > 0 vertical: shift 𝑓(𝑥) up

𝑦 = 𝑓(𝑥)−𝑘, 𝑘 > 0 : shift 𝑓(𝑥) down

• 𝑦 = 𝑓(𝑥 +ℎ) ℎ > 0 horizontal: shift 𝑓(𝑥) left

𝑦 = 𝑓(𝑥 −ℎ), ℎ > 0 : shift 𝑓(𝑥) right

Partial preview of the text

Download PreCalculus Exam Formula Sheets and more Cheat Sheet Calculus in PDF only on Docsity!

MATH 2412-PreCalculus

Exam Formula Sheets

 System of Equations and Matrices

 3 Matrix Row Operations:

Switch any two rows.
Multiply any row by a nonzero constant.
Add any constant-multiple row to another

 Even and Odd functions

• Even function: 𝑓(−𝑥) = 𝑓(𝑥) Odd function: 𝑓(−𝑥) = −𝑓(𝑥)

 Graph Symmetry

• 𝑥-axis symmetry: if (𝑥, 𝑦) is on the graph, then (𝑥, −𝑦) is also on the graph

• 𝑦-axis symmetry: if (𝑥, 𝑦) is on the graph, then (−𝑥, 𝑦) is also on the graph

• origin symmetry: if (𝑥, 𝑦)f is on the graph, then (−𝑥, −𝑦) is also on the graph

 Function Transformations

 Stretch and Compress

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

 Reflections

• 𝑦 = −𝑓(𝑥) reflect 𝑓(𝑥) about 𝑥-axis

• 𝑦 = 𝑓(−𝑥) reflect 𝑓(𝑥) about 𝑦-axis

 Stretch and Compress

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

: compress 𝑓(𝑥) if 0 < 𝑎 < 1

• 𝑦 = 𝑓(𝑎𝑥), 𝑎 > 0 horizontal: stretch 𝑓(𝑥) if 0 < 𝑎 < 1

: compress 𝑓(𝑥) if 𝑎 > 1

 Shifts

• 𝑦 = 𝑓(𝑥) + 𝑘, 𝑘 > 0 vertical: shift 𝑓(𝑥) up

𝑦 = 𝑓(𝑥) − 𝑘, 𝑘 > 0 : shift 𝑓(𝑥) down

• 𝑦 = 𝑓(𝑥 + ℎ) ℎ > 0 horizontal: shift 𝑓(𝑥) left

𝑦 = 𝑓(𝑥 − ℎ), ℎ > 0 : shift 𝑓(𝑥) right

MATH 2412-PreCalculus

Exam Formula Sheets

 Formulas/Equations

• Slope Intercept: 𝑦 = 𝑚𝑥 + 𝑏 Point-Slope: 𝑦 − 𝑦 1 = 𝑚(𝑥 − 𝑥 1 )

• Slope: 𝑚 = 𝑦 𝑥^2 −𝑦^1

2 −𝑥 1

Average Rate of Change: Δ𝑦Δ𝑥 = 𝑓(𝑏)−𝑓(𝑎)𝑏−𝑎 , where 𝑎 ≠ 𝑏

• Circle: 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋𝑟 = 𝜋𝑑, 𝐴𝑟𝑒𝑎 = 𝜋𝑟^2

• Triangle: 𝐴𝑟𝑒𝑎 = 12 𝑏ℎ

• Rectangle: 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑙 + 2𝑤 , 𝐴𝑟𝑒𝑎 = 𝑙𝑤

• Rectangular Solid: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑤ℎ, 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ

• Sphere: 𝑉𝑜𝑙𝑢𝑚𝑒 =

4

• Right Circular Cylinder: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟^2 ℎ , 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝜋𝑟^2 + 2𝜋𝑟ℎ

 General Form of Quadratic Function: 𝑓(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 , (𝑎 ≠ 0)

• Quadratic Formula: 𝑥 = −𝑏±√𝑏

(^2) −4𝑎𝑐 2𝑎

• Vertex (ℎ, 𝑘): ℎ = − 2𝑎𝑏 𝑘 = 𝑎(ℎ)^2 + 𝑏(ℎ) + 𝑐,

or (−

2𝑎 , 𝑓 (−^

2𝑎)),^ or^ (−^

2𝑎 ,^

4𝑎𝑐−𝑏^2

• Axis of symmetry: 𝑥 = ℎ

 Vertex Form of Quadratic Function: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)^2 + 𝑘 vertex (ℎ, 𝑘)

 Polynomial function: 𝑓(𝑥) = 𝑎𝑛𝑥𝑛^ + 𝑎𝑛−1𝑥𝑛−1^ + ⋯ + 𝑎 1 𝑥^1 + 𝑎 0

 Polynomial graph has at most 𝑛 − 1 turning points.

 Remainder Theorem

• If polynomial 𝑓(𝑥) ÷ (𝑥 − 𝑐), remainder is 𝑓(𝑐).

 Factor Theorem

• If 𝑓(𝑐) = 0, then 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥).

• If 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥), then 𝑓(𝑐) = 0.

MATH 2412-PreCalculus

Exam Formula Sheets

• If 𝑀 = 𝑁, then log𝑎(𝑀) = log𝑎(𝑁).

• Change of Base formula log𝑎(𝑀) = log(𝑀)log(𝑎) or log𝑎(𝑀) = ln(𝑀)ln(𝑎)

 Exponential Models Formulas

• Simple Interest: 𝐼 = 𝑃𝑟𝑡

• Compound Interest: 𝐴 = 𝑃 (1 + 𝑛𝑟)

𝑛∙𝑡

• Continuous Compounding: 𝐴 = 𝑃𝑒𝑟∙𝑡

Effective Rate of Interest:

Compounding 𝑛 times per year 𝑟𝑒𝑓𝑓 = (1 + 𝑛𝑟)

𝑛

Compounding continuously per year 𝑟𝑒𝑓𝑓 = 𝑒𝑟^ − 1

• Growth & Decay: 𝐴(𝑡) = 𝐴 0 𝑒𝑘∙𝑡

• Newton’s Law of Cooling: 𝑢(𝑡) = 𝑇 + (𝑢 0 − 𝑇)𝑒𝑘∙𝑡

• Logistic Model: 𝑃(𝑡) = 1+𝑎𝑒𝑐−𝑏∙𝑡

 Sequences and Series

𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) ∙ ⋯ ∙ (3)(2)(1)

Arithmetic Sequence:

𝑛𝑡ℎ^ term 𝑎𝑛 = 𝑎 1 + (𝑛 − 1)𝑑

Sum of first 𝑛 terms 𝑆𝑛 = ∑ 𝑛𝑘=1(𝑎 1 + (𝑘 − 1)𝑑)= 𝑛 2 (𝑎 1 + 𝑎𝑛)

or 𝑆𝑛 = ∑ 𝑛𝑘=1 (𝑎 1 + (𝑘 − 1)𝑑)= 𝑛 2 (2𝑎 1 + (𝑛 − 1)𝑑).

Geometric Sequence:

𝑛𝑡ℎ^ term 𝑎𝑛 = 𝑎 1 (𝑟)𝑛−

Sum of first 𝑛 terms 𝑆𝑛 = ∑ 𝑛𝑘=1 𝑎 1 𝑟𝑘−1= 𝑎 1 ∙ 1−𝑟

𝑛

1−𝑟 for^ 𝑟 ≠ 0,

• Geometric Series: ∑ ∞𝑘=1 𝑎 1 𝑟𝑘−1= 1−𝑟𝑎^1 if |𝑟| < 1

MATH 2412-PreCalculus

Exam Formula Sheets

 Binomial Theorem:

(𝑥 + 𝑎)𝑛^ = ∑ 𝑛𝑗=0 (𝑛𝑗 ) 𝑥𝑛−𝑗𝑎𝑗= (𝑛 0 )𝑥𝑛^ + (𝑛 1 )𝑥𝑛−1𝑎 + ⋯ + ( 𝑛−1𝑛 )𝑥𝑎𝑛−1^ + (𝑛𝑛)𝑎𝑛

 Trigonometry

 Circular Measure and Motion Formulas

Arc Length 𝑠 = 𝑟𝜃 Area of Sector 𝐴 = 12 𝑟^2 𝜃
Linear Speed 𝑣 = 𝑠𝑡 , 𝑣 = 𝑟𝜔 Angular Speed 𝜔 = 𝜃𝑡  Acute Angle
sin(𝜃) = 𝑏𝑐 = (^) ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 cos(𝜃) = 𝑎𝑐 = (^) ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan(𝜃) = 𝑏𝑎 = (^) 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
csc(𝜃) = 𝑐𝑏 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 sec(𝜃) = (^) 𝑎𝑐 = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 cot(𝜃) = 𝑎𝑏 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒  General Angle
sin(𝜃) = 𝑏𝑟 cos(𝜃) = 𝑎𝑟 tan(𝜃) = 𝑏𝑎
csc(𝜃) = 𝑟𝑏 ,𝑏 ≠ 0 sec(𝜃) = (^) 𝑎𝑟 ,𝑎 ≠ 0 cot(𝜃) = 𝑎𝑏 ,𝑏 ≠ 0  Cofunctions
sin(𝜃) = cos (𝜋 2 − 𝜃) , cos(𝜃) = sin (𝜋 2 − 𝜃) , tan(𝜃) = cot (𝜋 2 − 𝜃)
csc(𝜃) = sec (𝜋 2 − 𝜃) , sec(𝜃) = csc (𝜋 2 − 𝜃) , cot(𝜃) = tan (𝜋 2 − 𝜃)  Fundamental Identities
tan(𝜃) = sin(𝜃)cos(𝜃) cot(𝜃) = cos(𝜃)sin(𝜃)
csc(𝜃) = (^) sin(𝜃)^1 sec(𝜃) = (^) cos(𝜃)^1 cot(𝜃) = (^) tan(𝜃)^1
sin^2 (𝜃) + cos^2 (𝜃) = 1 tan^2 (𝜃) + 1 = sec^2 (𝜃)^ cot^2 (𝜃) + 1 = csc^2 (𝜃)  Even-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)  Inverse Functions
𝑦 = sin−1(𝑥) means 𝑥 = sin(𝑦) where −1 ≤ 𝑥 ≤ 1 and − 𝜋 2 ≤ 𝑦 ≤ 𝜋 2
𝑦 = cos−1(𝑥) means 𝑥 = cos(𝑦) where −1 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑦 ≤ 𝜋
𝑦 = tan−1(𝑥) means 𝑥 = tan(𝑦) where −∞ ≤ 𝑥 ≤ ∞ and − 𝜋 2 < 𝑦 < 𝜋 2

MATH 2412-PreCalculus

Exam Formula Sheets

sin(𝛼) − sin(𝛽) = 2 sin (𝛼−𝛽 2 ) cos (𝛼+𝛽 2 )
cos(𝛼) + cos(𝛽) = 2 cos (𝛼+𝛽 2 ) cos (𝛼−𝛽 2 )
cos(𝛼) − cos(𝛽) = −2 sin (𝛼+𝛽 2 ) sin (𝛼−𝛽 2 )

 Law of Sines

sin(𝐴)𝑎 = sin(𝐵)𝑏 = sin(𝐶)𝑐

 Law of Cosines

𝑎^2 = 𝑏^2 + 𝑐^2 − 2𝑏𝑐 cos(𝐴)
𝑏^2 = 𝑎^2 + 𝑐^2 − 2𝑎𝑐 cos(𝐵)
𝑐^2 = 𝑎^2 + 𝑏^2 − 2𝑎𝑏 cos(𝐶)

 Area of SSS Triangles (Heron’s Formula)

𝐾 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) , where 𝑠 = 12 (𝑎 + 𝑏 + 𝑐)

 Area of SAS Triangles

𝐾 = 12 𝑎𝑏 sin(𝐶) , 𝐾 = 12 𝑏𝑐 sin(𝐴) , 𝐾 = 12 𝑎𝑐 sin(𝐵)

PreCalculus Exam Formula Sheets, Cheat Sheet of Calculus

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Partial preview of the text

Download PreCalculus Exam Formula Sheets and more Cheat Sheet Calculus in PDF only on Docsity!

MATH 2412-PreCalculus

Exam Formula Sheets

• Even function: 𝑓(−𝑥) = 𝑓(𝑥) Odd function: 𝑓(−𝑥) = −𝑓(𝑥)

• 𝑥-axis symmetry: if (𝑥, 𝑦) is on the graph, then (𝑥, −𝑦) is also on the graph

• 𝑦-axis symmetry: if (𝑥, 𝑦) is on the graph, then (−𝑥, 𝑦) is also on the graph

• origin symmetry: if (𝑥, 𝑦)f is on the graph, then (−𝑥, −𝑦) is also on the graph

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

• 𝑦 = −𝑓(𝑥) reflect 𝑓(𝑥) about 𝑥-axis

• 𝑦 = 𝑓(−𝑥) reflect 𝑓(𝑥) about 𝑦-axis

• 𝑦 = 𝑎𝑓(𝑥), 𝑎 > 0 vertical: stretch 𝑓(𝑥) if 𝑎 > 1

: compress 𝑓(𝑥) if 0 < 𝑎 < 1

• 𝑦 = 𝑓(𝑎𝑥), 𝑎 > 0 horizontal: stretch 𝑓(𝑥) if 0 < 𝑎 < 1

: compress 𝑓(𝑥) if 𝑎 > 1

• 𝑦 = 𝑓(𝑥) + 𝑘, 𝑘 > 0 vertical: shift 𝑓(𝑥) up

𝑦 = 𝑓(𝑥) − 𝑘, 𝑘 > 0 : shift 𝑓(𝑥) down

• 𝑦 = 𝑓(𝑥 + ℎ) ℎ > 0 horizontal: shift 𝑓(𝑥) left

𝑦 = 𝑓(𝑥 − ℎ), ℎ > 0 : shift 𝑓(𝑥) right

MATH 2412-PreCalculus

Exam Formula Sheets

• Slope Intercept: 𝑦 = 𝑚𝑥 + 𝑏 Point-Slope: 𝑦 − 𝑦 1 = 𝑚(𝑥 − 𝑥 1 )

• Slope: 𝑚 = 𝑦 𝑥^2 −𝑦^1

• Circle: 𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋𝑟 = 𝜋𝑑, 𝐴𝑟𝑒𝑎 = 𝜋𝑟^2

• Triangle: 𝐴𝑟𝑒𝑎 = 12 𝑏ℎ

• Rectangle: 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝑙 + 2𝑤 , 𝐴𝑟𝑒𝑎 = 𝑙𝑤

• Rectangular Solid: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑤ℎ, 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ

• Sphere: 𝑉𝑜𝑙𝑢𝑚𝑒 =

• Right Circular Cylinder: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟^2 ℎ , 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2𝜋𝑟^2 + 2𝜋𝑟ℎ

 General Form of Quadratic Function: 𝑓(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 , (𝑎 ≠ 0)

• Quadratic Formula: 𝑥 = −𝑏±√𝑏

• Vertex (ℎ, 𝑘): ℎ = − 2𝑎𝑏 𝑘 = 𝑎(ℎ)^2 + 𝑏(ℎ) + 𝑐,

or (−

2𝑎 , 𝑓 (−^

2𝑎)),^ or^ (−^

2𝑎 ,^

4𝑎𝑐−𝑏^2

• Axis of symmetry: 𝑥 = ℎ

 Vertex Form of Quadratic Function: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)^2 + 𝑘 vertex (ℎ, 𝑘)

 Polynomial function: 𝑓(𝑥) = 𝑎𝑛𝑥𝑛^ + 𝑎𝑛−1𝑥𝑛−1^ + ⋯ + 𝑎 1 𝑥^1 + 𝑎 0

 Polynomial graph has at most 𝑛 − 1 turning points.

• If polynomial 𝑓(𝑥) ÷ (𝑥 − 𝑐), remainder is 𝑓(𝑐).

• If 𝑓(𝑐) = 0, then 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥).

• If 𝑥 − 𝑐 is a linear factor of 𝑓(𝑥), then 𝑓(𝑐) = 0.

MATH 2412-PreCalculus

Exam Formula Sheets

• If 𝑀 = 𝑁, then log𝑎(𝑀) = log𝑎(𝑁).

• Change of Base formula log𝑎(𝑀) = log(𝑀)log(𝑎) or log𝑎(𝑀) = ln(𝑀)ln(𝑎)

• Simple Interest: 𝐼 = 𝑃𝑟𝑡

• Compound Interest: 𝐴 = 𝑃 (1 + 𝑛𝑟)

• Continuous Compounding: 𝐴 = 𝑃𝑒𝑟∙𝑡

Compounding 𝑛 times per year 𝑟𝑒𝑓𝑓 = (1 + 𝑛𝑟)

Compounding continuously per year 𝑟𝑒𝑓𝑓 = 𝑒𝑟^ − 1

• Growth & Decay: 𝐴(𝑡) = 𝐴 0 𝑒𝑘∙𝑡

• Newton’s Law of Cooling: 𝑢(𝑡) = 𝑇 + (𝑢 0 − 𝑇)𝑒𝑘∙𝑡

• Logistic Model: 𝑃(𝑡) = 1+𝑎𝑒𝑐−𝑏∙𝑡

𝑛𝑡ℎ^ term 𝑎𝑛 = 𝑎 1 + (𝑛 − 1)𝑑

Sum of first 𝑛 terms 𝑆𝑛 = ∑ 𝑛𝑘=1(𝑎 1 + (𝑘 − 1)𝑑)= 𝑛 2 (𝑎 1 + 𝑎𝑛)

or 𝑆𝑛 = ∑ 𝑛𝑘=1 (𝑎 1 + (𝑘 − 1)𝑑)= 𝑛 2 (2𝑎 1 + (𝑛 − 1)𝑑).

𝑛𝑡ℎ^ term 𝑎𝑛 = 𝑎 1 (𝑟)𝑛−

Sum of first 𝑛 terms 𝑆𝑛 = ∑ 𝑛𝑘=1 𝑎 1 𝑟𝑘−1= 𝑎 1 ∙ 1−𝑟

1−𝑟 for^ 𝑟 ≠ 0,

• Geometric Series: ∑ ∞𝑘=1 𝑎 1 𝑟𝑘−1= 1−𝑟𝑎^1 if |𝑟| < 1

MATH 2412-PreCalculus

Exam Formula Sheets

(𝑥 + 𝑎)𝑛^ = ∑ 𝑛𝑗=0 (𝑛𝑗 ) 𝑥𝑛−𝑗𝑎𝑗= (𝑛 0 )𝑥𝑛^ + (𝑛 1 )𝑥𝑛−1𝑎 + ⋯ + ( 𝑛−1𝑛 )𝑥𝑎𝑛−1^ + (𝑛𝑛)𝑎𝑛

MATH 2412-PreCalculus

Exam Formula Sheets

 For 𝑦 = 𝐴sin(𝜔𝑥 − 𝜑) or 𝑦 = 𝐴cos(𝜔𝑥 − 𝜑) , with 𝜔 > 0

• Amplitude = |𝐴| , Period= 𝑇 = 2𝜋𝜔 , Phase shift = 𝜑𝜔