Trigonometry Basics (Tangent, Sine, Cosine Functions)

Each angle of a right-angled triangle has a constant ratio called the tangent.

Angle Notation

ABC\text{ABC} \Rarr The angle at point BB.
BAC\text{BAC} \Rarr The angle at point AA.

Or just B\angle B, B^\hat{B} or A\angle A, A^\hat{A}

Lowercase greek letters are also commonly used to denote angles:

α\alpha alpha
β\beta beta
θ\theta theta
ϕ\phi phi

Tangent Function

The tangent function of an angle is equal to the ratio of the angle's opposite and adjacent sides.

Given θ°=BAC\theta\degree = BAC and ϕ°=ABC\phi\degree = ABC

Applicable angles are either of the two angles that are not 90°90\degree.


tanθ°=hb\tan \theta\degree = \frac h b
tanϕ°=bh\tan \phi\degree = \frac b h

Inverse Tangent Function

The tangent function's inverse calculates an angle based on a known ratio.

tan1hb=θ°\tan^{-1} \frac h b = \theta\degree


All angles of a right-angled triangle always sum up to 180°180\degree. Since the right-angle is always one of 90°90\degree, the remaining angles always add up to another 90°90\degree. Therefore:

90°θ°=ϕ°90\degree - \theta\degree = \phi\degree and
90°ϕ°=θ°90\degree - \phi\degree = \theta\degree

tan45°=1\tan 45\degree = 1
tan90°=invalid\tan 90\degree = \text{invalid}

The angle being worked on is called the angle of focus. That angle's tangent ratio is always oppositeadjacent\frac{\text{opposite}}{\text{adjacent}}

Sine and Cosine Functions

sinθ°=oppositehypotenuse\sin \theta\degree = \frac{\text{opposite}}{\text{hypotenuse}}
cosθ°=adjacenthypotenuse\cos \theta\degree = \frac{\text{adjacent}}{\text{hypotenuse}}

The inverse functions sin1\sin^{-1} and cos1\cos^{-1} yield the angle given a ratio.

Non-right-angled Triangles

Any angle with a perpendicular opposite can be found by bisecting the opposite at that angle by creating two right-angled triangles.

Pythagorean Identity and Theorem

cos2θ+sin2θ=1\cos^2\theta + \sin^2\theta = 1

a2+b2=c2a^2 + b^2 = c^2 where aa and bb are equal to the lengths of the adjacent and opposite, and cc is equal to the length of the hypotenuse.

Relationship of the Functions

Since sinθ°=y\sin \theta\degree = y, and cosθ°=x\cos \theta\degree = x, and
tanθ°=yx\tan \theta\degree = \frac y x, it follows that:

tanθ°=sinθ°cosθ°\tan \theta\degree = \frac {\sin \theta\degree}{\cos \theta\degree}