All The Math That You Need To Start Doing Quantum Computing (Part-2)

Lesson 2: Trigonometry

Here’s the second part of the series that introduces all the mathematics you must understand to start doing quantum mechanics and quantum computing.

Here’s the first one if you missed it.

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Lesson 2: Trigonometry

Trigonometric functions define the relationships between angles and the ratio of lengths of a right-angled triangle.

Consider the right-angled triangle shown below:

Right-angled triangle (Credits: Tarquin & Limaner, Wikimedia)

The trigonometric functions associated with this triangle are as follows:

The reciprocal to these functions are as follows:

where:

• sin is sine

• cos is cosine

• tan is tangent

• csc is cosecant

• sec is secant

• cot is cotangent

Note that the ratios can be remembered using the mnemonic:

SOH-CAH-TOA

where:

• Sine = Opposite ÷ Hypotenuse

• Cosine = Adjacent ÷ Hypotenuse

• Tangent = Opposite ÷ Adjacent
  

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Plots of Trigonometric Functions

The plot of sin(x) is shown below:

Image from Wikipedia By Geek3

The plot of cos(x) is shown below:

Image from Wikipedia by Geek3

Note that when we compare the plots of sin(x) and cos(x), they are at a phase difference of π.

This means that if we move the curve of cos(x) relative to sin(x) by π, both will overlap.

The plot of tan(x) is shown below:

Image from Wikipedia by Geek3

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Calculating Values of Trigonometric Functions

If we draw a right-angled triangle inside a circle of radius of unit length, the position of the point where the radius intersects the circumference of the circle can be represented by (x, y) where:

x = cos θ

y = sin θ

Image from Wikipedia by By 345Kai

Values of trigonometric functions for different θs can be found by drawing a triangle inside a circle and calculating its ratios.

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Radian

Angles, as described above, are commonly measured in Radians.

One radian is the angle subtended from the centre of a circle by an arc equal to the circle's radius.

Image from Wikipedia By Lucas Vieira

2π radians = 360°

1 radian = 57.2958°

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Trigonometric identities

• The following identities are Pythagoras Theorem in disguise.

  

• Below-mentioned identities are Euler’s Formula in disguise.

• If an angle θ can be written as a sum or difference of two other angles α and β i.e. θ = α ± β

  

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I hope this article helped you grasp some Trigonometric concepts, as we will use them extensively in Quantum computing.

This list is not exhaustive but good enough to start with.

See you soon in the next part.