# Trig Ratios

• Sep 19th 2012, 06:04 PM
Oldspice1212
Trig Ratios
Find the remaining trigonometric ratios.

tan (a) = 4, 0<a<pi/2

 sin(α) = cos(α) = cot(α) = sec(α) = csc(α) =
• Sep 19th 2012, 06:36 PM
MarkFL
Re: Trig Ratios
I would use the Pythagorean identities, keeping in mind all roots will be positive since the angle given is in the first quadrant. For example:

$\displaystyle \tan(a)=4$

$\displaystyle \frac{\sin(a)}{\cos(a)}=4$

$\displaystyle \sin(a)=4\cos(a)$

$\displaystyle \sin^2(a)=16\cos^2(a)$

$\displaystyle \sin^2(a)=16(1-\sin^2(a))$

$\displaystyle 17\sin^2(a)=16$

$\displaystyle \sin^2(a)=\frac{16}{17}$

$\displaystyle \sin(a)=\frac{4}{\sqrt{17}}$

You can now easily find $\displaystyle \cos(a)$ from $\displaystyle \sin(a)=4\cos(a)$. Now that you have the three primary functions, finding the rest is straightforward...
• Sep 19th 2012, 06:43 PM
Oldspice1212
Re: Trig Ratios
So would cos(a) be the same as sin (a), and the rest would be just reciprocals? Sorry I haven't done this in a long time :S
• Sep 19th 2012, 06:52 PM
MarkFL
Re: Trig Ratios
No, from the equation relating sin(a) and cos(a), we see that cos(a) is 1/4 the value of sin(a). Then yes, the secondary functions would be the reciprocals of the corresponding primary functions.
• Sep 19th 2012, 09:00 PM
Soroban
Re: Trig Ratios
Hello, Oldspice1212!

Quote:

Given: .$\displaystyle \tan\theta \,=\,4,\;0 < \theta < \tfrac{\pi}{2}$
Find the remaining trigonometric ratios.

We are given: .$\displaystyle \tan\theta\:=\:\frac{4}{1}\:=\:\frac{opp}{adj}$

$\displaystyle \theta$ is in a right triangle with: $\displaystyle opp = 4,\:adj = 1$
Pythagorus tells us that: $\displaystyle hyp \:=\:\sqrt{4^2+1^2} \:=\:\sqrt{17}$

So we have: .$\displaystyle \begin{Bmatrix}opp &=& 4 \\ adj &=& 1 \\ hyp &=& \sqrt{17} \end{Bmatrix}$

And we can write all six trig ratios:

. . $\displaystyle \begin{array}{cccccccccc}\sin\theta &=& \frac{4}{\sqrt{17}} &\;\;& \csc\theta &=& \frac{\sqrt{17}}{4} \\ \\[-3mm] \cos\theta &=& \frac{1}{\sqrt{17}} && \sec\theta &=& \sqrt{17} \\ \\[-3mm] \tan\theta &=& 4 && \cot\theta &=& \frac{1}{4} \end{array}$