Question: "Why does tan equal to sin/cos?"

gibbokf5

Answered question

2022-12-30

Why does tan equal to sin/cos?

Gremoli9bs

Beginner2022-12-31Added 10 answers

Unit circle:

t

correspond to point

(x, y)

on the circle

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

Define:

\sin t = y

, ,

\cos t = x

, ,

\tan t = \frac{y}{x}

The outcome occurs instantly.
Point

(x, y)

on the terminal side of

t

and

r = \sqrt{x^{2} + y^{2}}

\sin t = \frac{y}{r}

, ,

\cos t = \frac{x}{r}

, ,

\tan t = \frac{y}{x}

\frac{\sin t}{\cos t} = \frac{\frac{y}{r}}{\frac{x}{r}} = (\frac{y}{r}) \cdot (\frac{r}{x}) = \frac{y}{x} = \tan t

Right triangle

\sin θ = o p p o s i t e \frac{}{} h y p o t e ν s e = o p p \frac{}{} h y p

\cos θ = a d j a c e n t \frac{}{} h y p o t e ν s e = a d j \frac{}{} h y p

\sin θ = o p p o s i t e \frac{}{} a d j a c e n t = o p p \frac{}{} a d j

The result follow from :

\frac{\sin θ}{\cos θ} = \frac{o p p \frac{}{} h y p}{a d j \frac{}{} h y p} = (o p p \frac{}{} h y p) \cdot (h y p \frac{}{} a d j) = o p p \frac{}{} a d j = \tan θ

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