1
$\begingroup$

As the cosine is defined as $cos \theta = \frac x r $ where x is the length of opposite line and r is length of hypotenuse how can $cos \theta$ be calculated for values where we do not know the lengths of sides of triangles. ?

Various calculators compute the $cos(1) = .999$ , so is there an implicit value for lengths of opposite and adjacent sides ?

  • 0
    Make a right-angled triangle with the other angles as $1,89$. Then, make one of either the opposite or adjacent side lengths as $1$. Now see how $\cos 1$ is calculated. – 2017-02-22
  • 0
    Suppose you have another right triangle with angle $\theta$, opposite side $x'$ and hypothenuse $r'$. Then it is similar to the other triangle, in particular $\dfrac{x}{r}=\dfrac{x'}{r'}$, so it's irrelevant what “implicit length” you choose. – 2017-02-22
  • 0
    @egreg insight I had been missing is that $\theta$ remains constant between right angled triangles if they are proportionally equivalent. thanks. – 2017-02-22

2 Answers 2

1

This picture should solve your doubt:

enter image description here

Say the angle $\widehat{BAC}$ is $\theta$; then $$ \cos\theta=\frac{b}{r}=\frac{b'}{r'} $$ where $b=AB$, $b'=AD$, $r=AC$ and $r'=AE$, because the two triangles $ABC$ and $ADE$ are similar right triangles.

So, while one needs an “implicit value” for the length, it's immaterial which one you choose, because the ratio will be the same.

In the past, in order to avoid decimal numbers, the length of the hypothenuse was taken to be, say, $100\,000$ and the cosine was defined to be the adjacent leg (no ratio). However, this just complicates formulas, so it became common to use an implicit length of $1$ for the hypothenuse, which is the same as taking the ratio.

Thus, saying that $\cos 1^\circ\approx 0.9998$ means that if the hypothenuse is $1$ (meter, foot, yard, centimeter), then the adjacent leg will be approximately $0.9998$ (in meters, feet, yards, centimeters, whatever unit you chose). If the hypothenuse is $2$, you'd get an adjacent leg approximately $1.9996$.

1

As

$$\frac xr=\frac{\lambda x}{\lambda r}$$ you can take any value for one of the sides and the other varies in proportion.

The implicit value is always taken to be $1$ for the length of the hypothenuses.

  • 0
    can you elaborate on $\frac xr=\frac{\lambda x}{\lambda r}$ ? What do the $\lambda$ values represent ? Just knowing the length of the hypothenuses is not sufficient to calculate cosine as also require length of opposite side ? – 2017-02-22