Trigonometry & Complex Numbers

Trigonometry - Geometric View

Geometric view of trigonometric functions

Function	Syntax	Description		Relationship	Relationships
					Using Radians	Using Degrees
sine	𝑠𝑖𝑛(𝜃)	opposite / hypotenuse	SOH	if hypotenuse = 1, then it’s just opposite	$sin(𝜃)=cos(\frac{𝜋}{2}-𝜃)=\frac{1}{csc(𝜃)}$	$sin(x)=cos(90°-x)=\frac{1}{csc(x)}$
cosine	𝑐𝑜𝑠(𝜃)	adjacent / hypotenuse	CAH	if hypotenuse = 1, then it’s just adjacent	$cos(𝜃)=sin(\frac{𝜋}{2}-𝜃)=\frac{1}{sec(𝜃)}$	$cos(x)=sin(90°-x)=\frac{1}{sec(x)}$
tangent	𝑡𝑎𝑛(𝜃)	opposite / adjacent	TOA	if adjacent = 1, then it’s just opposite	$tan(𝜃)=\frac{sin(𝜃)}{cos(𝜃)}=cot(\frac{𝜋}{2}-𝜃)=\frac{1}{cot(𝜃)}$	$tan(x)=\frac{sin(x)}{cos(x)}=cot(90°-x)=\frac{1}{cot(x)}$
cosecant	𝑐𝑠𝑐(𝜃)	hypotenuse / opposite	CO SEC HO	if opposite = 1, then it’s just hypotenuse	$csc(𝜃)=sec(\frac{𝜋}{2}-𝜃)=\frac{1}{sin(𝜃)}$	$csc(x)=sec(90°-x)=\frac{1}{sin(x)}$
secant	𝑠𝑒𝑐(𝜃)	hypotenuse / adjacent	SEC HA	if adjacent = 1, then it’s just hypotenuse	$sec(𝜃)=csc(\frac{𝜋}{2}-𝜃)=\frac{1}{cos(𝜃)}$	$sec(x)=csc(90°-x)=\frac{1}{cos(x)}$
cotangent	𝑐𝑜𝑡(𝜃)	adjacent / opposite	CO TAO	if opposite = 1, then it’s just adjacent	$cot(𝜃)=\frac{cos(𝜃)}{sin(𝜃)}=tan(\frac{𝜋}{2}-𝜃)=\frac{1}{tan(𝜃)}$	$cot(x)=\frac{cos(x)}{sin(x)}=tan(90°-x)=\frac{1}{tan(x)}$
cosine squared	𝑐𝑜𝑠2(𝜃)	𝑐𝑜𝑠2(𝜃) is running 𝑐𝑜𝑠(𝜃) twice. For example: 𝑐𝑜𝑠(𝜃) = adjacent / hypotenuse if hypotenuse = 𝑐𝑜𝑠(𝜃), then: 𝑐𝑜𝑠(𝜃) = adjacent / 𝑐𝑜𝑠(𝜃) thus, 𝑐𝑜𝑠2(𝜃) = adjacent		𝑐𝑜𝑠2(𝜃) is the area of the square with side 𝑐𝑜𝑠(𝜃)	1 = 𝑠𝑖𝑛2(𝜃) + 𝑐𝑜𝑠2(𝜃)	trigonometry-geometric-view.drawio
sine squared	𝑠𝑖𝑛2(𝜃)	𝑠𝑖𝑛2(𝜃) is running 𝑠𝑖𝑛(𝜃) twice. For example: 𝑠𝑖𝑛(𝜃) = opposite / hypotenuse if hypotenuse = 𝑠𝑖𝑛(𝜃), then: 𝑠𝑖𝑛(𝜃) = opposite / 𝑠𝑖𝑛(𝜃) thus, 𝑠𝑖𝑛2(𝜃) = opposite		𝑠𝑖𝑛2(𝜃) is the area of the square with side 𝑠𝑖𝑛(𝜃)

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Complex Numbers

• adding complex numbers is simple
• multiplying complex numbers is like rotating and scaling. For example, multiplying a complex number by 𝑖 is like rotating it 90 degrees counterclockwise

Computing 𝑐𝑜𝑠(75°)

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𝑐𝑜𝑠(75) = Is like taking [1,0] and rotating it 45 degrees and then 30 degrees counter-clockwise

𝑐𝑜𝑠(75) = [1, 0] [𝑐𝑜𝑠(45), 𝑖·𝑠𝑖𝑛(45)] [𝑐𝑜𝑠(30), 𝑖·𝑠𝑖𝑛(30)]

• 𝑐𝑜𝑠(45) = √(1/2)
• 𝑠𝑖𝑛(45) = √(1/2)
• 𝑐𝑜𝑠(30) = √(3)/2
• 𝑠𝑖𝑛(30) = 1/2

𝑐𝑜𝑠(75) = [1, 0] [√(1/2), i√(1/2)] [√(3)/2, i(1/2)]
              = [1, 0] [√(1/2)*√(3)/2 - √(1/2)(1/2), √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [√(1/2)*√(3)(1/2) - √(1/2)(1/2), √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [√(3/2)*√(1/4) - √(1/2)√(1/4), √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [√(6/4)*√(1/4) - √(2/4)√(1/4), √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [√(6/16) - √(2/16), √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [√(6)/4 - √(2)/4, √(1/2)√(3)/2 + √(1/2)/2]
              = [1, 0] [(1/4)[√(6) - √(2)], √(1/2)√(3)/2 + √(1/2)/2]
              = [(1/4)[√(6) - √(2)], 0]
              = (1/4)[√(6) - √(2)]

Multiplying Complex Numbers is Like Adding Rotations Together

1. [𝑐𝑜𝑠(𝑥) + 𝑖·𝑠𝑖𝑛(𝑥)] * [𝑐𝑜𝑠(𝑦) + 𝑖·𝑠𝑖𝑛(𝑦)] = [𝑐𝑜𝑠(x+y) + 𝑖·𝑠𝑖𝑛(x+y)]
2. [𝑐𝑜𝑠(𝑥) + 𝑖·𝑠𝑖𝑛(𝑥)] * [𝑐𝑜𝑠(𝑦) + 𝑖·𝑠𝑖𝑛(𝑦)] = [𝑐𝑜𝑠(𝑥)𝑐𝑜𝑠(𝑦) - 𝑠𝑖𝑛(𝑥)𝑠𝑖𝑛(𝑦) + i𝑐𝑜𝑠(𝑦)𝑠𝑖𝑛(𝑥) + i𝑐𝑜𝑠(𝑥)𝑠𝑖𝑛(𝑦)]
3. [𝑐𝑜𝑠(𝑥) + 𝑖·𝑠𝑖𝑛(𝑥)] * [𝑐𝑜𝑠(𝑦) + 𝑖·𝑠𝑖𝑛(𝑦)] = [𝑐𝑜𝑠(𝑥)𝑐𝑜𝑠(𝑦) - 𝑠𝑖𝑛(𝑥)𝑠𝑖𝑛(𝑦) + i [𝑐𝑜𝑠(𝑦)𝑠𝑖𝑛(𝑥) + 𝑐𝑜𝑠(𝑥)𝑠𝑖𝑛(𝑦)]]

Because of 1 and 3 above, we have:

1. 𝑐𝑜𝑠(𝑥+𝑦) = 𝑐𝑜𝑠(𝑥)𝑐𝑜𝑠(𝑦) - 𝑠𝑖𝑛(𝑥)𝑠𝑖𝑛(𝑦)
2. 𝑠𝑖𝑛(𝑥+𝑦) = 𝑐𝑜𝑠(𝑦)𝑠𝑖𝑛(𝑥) + 𝑐𝑜𝑠(𝑥)𝑠𝑖𝑛(𝑦)

Because of 1 above, we have:

• 𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠²(𝑥) - 𝑠𝑖𝑛²(𝑥)
• 𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠²(𝑥) - (1 - 𝑐𝑜𝑠²(𝑥)) # because 1 = 𝑐𝑜𝑠²(𝑥) + 𝑠𝑖𝑛²(𝑥)
• 𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠²(𝑥) - (1 - 𝑐𝑜𝑠²(𝑥))
• 𝑐𝑜𝑠(2𝑥) = 2𝑐𝑜𝑠²(𝑥) - 1

Thus:

• 𝑐𝑜𝑠²(𝑥) = [𝑐𝑜𝑠(2𝑥) + 1] / 2

Euler’s Formula

• 𝑒^𝑖𝑥 = 𝑐𝑜𝑠(𝑥) + 𝑖·𝑠𝑖𝑛(𝑥)

see: Euler’s Formula - Intuition via Trigonometry & Complex Numbers

Resources

• #1 Trigonometry Fundamentals - 3Brown1Blue
• #2 Complex Number Fundamentals - 3Brown1Blue