Given tan(theta) = 8, find cos(theta) and sin(theta)
I have no idea how to do this lol, i forgot >_< and i want exact answers plz >_< so no decimals 0_o
think of the right triangle.
$\displaystyle \cos\theta=\frac{adjacent}{hypotenuse}$
$\displaystyle \sin\theta=\frac{opposite}{hypotenuse}$
$\displaystyle \tan\theta=\frac{opposite}{adjacent}$
Also Pythagorus' theorem: $\displaystyle a^2=b^2+c^2$
where a is the hypotenuse, and b and c are adjacent and opposite sides respectively.
Essentially the same: think of $\displaystyle tan(\theta)= 8$ as representing a right triangle with "near side" of length 8 and "opposite side" of length 1 so that $\displaystyle tan(\theta)$= "near side"/"opposite side"= 8. Use the Pythagorean theorem to find the hypotenuse and then use the formulas arze gives.
You haven't defined your domain, so you may also consider $\displaystyle Tan\theta$ to be the slope of a line through the origin, for $\displaystyle 0\ \le\ \theta\ \le\ 2{\pi}$.
Hence, there are also negative solutions for $\displaystyle Sin\theta$ and $\displaystyle Cos\theta$.
You don't know the lengths of the sides of your right-angled triangle, but all you need is the ratios.
$\displaystyle Adjacent=x$
$\displaystyle Opposite=8x$
$\displaystyle Hypotenuse=\sqrt{(8x)^2+x^2}=x\sqrt{65}$
Hello, haddad287!
$\displaystyle \text{Given: }\:\tan\theta \,=\, 8$
$\displaystyle \text{Find: }\;\cos\theta\,\text{ and }\,\sin\theta$
I will assume that $\displaystyle \,\theta$ is an acute angle.
$\displaystyle \text{We have: }\;\tan\theta \:=\:\dfrac{8}{1} \:=\:\dfrac{opp}{adj}$
That is: .$\displaystyle \,\theta$ is in a right triangle with $\displaystyle opp = 8,\;adj = 1$
. . (Can you picture that? If not, make a sketch.)
Pythagorus tell us that: .$\displaystyle hyp \:=\:\sqrt{8^2+1^2} \:=\:\sqrt{65}$
Therefore: .$\displaystyle \begin{Bmatrix}\cos\theta &=& \dfrac{adj}{hyp} &=& \dfrac{1}{\sqrt{65}} \\ \\[-3mm] \sin\theta &=& \dfrac{opp}{hyp} &=& \dfrac{8}{\sqrt{65}} \end{Bmatrix}$
Here is a different approach.
$\displaystyle
\tan\theta= \frac{\sin\theta}{\cos\theta} =8$
$\displaystyle \tan^2\theta=\frac{\sin^2\theta}{\cos^2\theta}=64$
$\displaystyle \sin^2\theta=64 \cos^2\theta$
$\displaystyle \sin^2\theta =64 (1-\sin^2\theta)$
$\displaystyle \sin^2\theta =64-64\sin^2\theta$
$\displaystyle 65\sin^2\theta = 64$
$\displaystyle \sin^2\theta = \frac{64}{65}$
$\displaystyle \sin\theta = \frac{8}{\sqrt{65}}$
$\displaystyle
\tan\theta= \frac{\sin\theta}{\cos\theta} =8$
$\displaystyle \tan^2\theta=\frac{\sin^2\theta}{\cos^2\theta}=64$
$\displaystyle \frac{1-\cos^2\theta}{\cos^2\theta}=64$
$\displaystyle
\frac{1}{\cos^2\theta}-\frac{\cos^2\theta}{\cos^2\theta}=64$
$\displaystyle \frac{1}{\cos^2\theta}-1=64$
$\displaystyle \frac{1}{\cos^2\theta}=65$
$\displaystyle \cos^2\theta=\frac{1}{65}$
$\displaystyle \cos\theta=\frac{1}{\sqrt{65}}$