Trigonometry One Page

We will take advantage of HTML's new (as of 2023) <math> tag to represent some useful fundamentals of trigonometry.

A look at the unit circle

\sin (- θ) - = \sin (θ)

\cos (- θ) = \cos (θ)

\tan (- θ) = - \tan (θ)

\sin (π - θ) = \sin (θ)

\cos (π - θ) = - \cos (θ)

\tan (π - θ) = - \tan (θ)

\sin (π + θ) = - \sin (θ)

\cos (π + θ) = - \cos (θ)

\tan (π + θ) = \tan (θ)

\sin (0 °) = 0

\cos (0 °) = 1

\tan (0 °) = 0

\sin (90 °) = 1

\cos (90 °) = 0

\tan (90 °) = undefined *

*i.e. "domain error", or "Invalid input", etc.

>>> import math
>>> math.tan(math.radians(90))
1.633123935319537e+16
>>>

sin(30°) = \frac{1}{2}

cos(30°) = \frac{\sqrt{3}}{2}

tan(30°) = \frac{1}{\sqrt{3}}

sin(45°) = \frac{\sqrt{2}}{2}

cos(45°) = \frac{\sqrt{2}}{2}

tan(45°) = 1

sin(60°) = \frac{\sqrt{3}}{2}

cos(60°) = \frac{1}{2}

tan(60°) = \sqrt{3}

Trigonometric Identities from the Unit Circle

tanθ = \frac{sinθ}{cosθ}

x^{2} + y^{2} = 1

\cos^{2} θ + \sin^{2} θ = 1

\cos^{2} θ = 1 - \sin^{2} θ

\sin^{2} θ = 1 - \cos^{2} θ

1 + \frac{\sin^{2} θ}{\cos^{2} θ} = \frac{1}{\cos^{2} θ}

1 + \tan^{2} θ = \sec^{2} θ

\frac{\cos^{2} θ}{\sin^{2} θ} + 1 = \frac{1}{\sin^{2} θ}

\cot^{2} θ + 1 = \csc^{2} θ

Complementary Angles

cosin = sin of the complementary angle

cotan = tan of the complementary angle

cosec = sec of the complementary angle

\cos θ = \sin (90 ° - θ)

\cot θ = \tan (90 ° - θ)

\csc θ = \sec (90 ° - θ)

\sin θ = \cos (90 ° - θ)

\tan θ = \cot (90 ° - θ)

\sec θ = \csc (90 ° - θ)

\begin{matrix} \cot θ & = & \tan (90 ° - θ) \\ = & \frac{\sin (90 ° - θ)}{\cos (90 ° - θ)} \\ = & \frac{\cos θ}{\sin θ} \\ = & 1 \div \frac{\sin θ}{\cos θ} \\ = & 1 \div \tan θ \end{matrix}

Example

Solve: \cos θ + \sin^{2} θ = 1

\begin{matrix} \cos θ + \sin^{2} θ & = & 1 \\ x^{2} + y^{2} & = & 1 \\ \cos^{2} θ + \sin^{2} θ & = & 1 \\ \sin^{2} θ & = & 1 - \cos^{2} θ \\ \cos θ + 1 - \cos^{2} θ & = & 1 \\ \cos θ - \cos^{2} θ & = & 0 \\ \cos θ (1 - \cos θ) & = & 0 \end{matrix}

\cos θ = 0 or 1 - \cos θ = 0

Lessons Learned

The following holds true both for mathematics in particular and for life in general. You may ask a question, and the answer that you receive does not satisfy your curiosity. The first answer that you receive — in practice — does not answer your question. So in this case, you should go and find someone else with a different perspective, and then ask your question again.