Tan theta equals 8 / 15 find sine theta + cos theta / cos theta (1 - cos theta)

Question

asked Mar 16, 2021 128k views

1 Answer

Hieu Nguyen Trung · Answer 1 · 2021-03-23T14:33:32+0000

I guess you have to find

$(\sin\theta+\cos\theta)/(\cos\theta(1-\cos\theta))$

given that
$\tan\theta=\frac8{15}$ .

We can immediately solve for
$\sec\theta$ :

$\sec^2\theta=1+\tan^2\theta\implies\sec\theta=\pm(17)/(15)$

(without knowing anything else about
$\theta$ , we cannot determine the sign)

Then we get
$\cos\theta$ for free:

$\cos\theta=\frac1{\sec\theta}=\pm(15)/(17)$

and we can now solve for
$\sin\theta$ :

$\sin^2\theta+\cos^2\theta=1\implies \sin\theta=\pm\frac8{17}$

Notice that we have 2*2 = 4 possible choices of sign for either sin or cos.

• If both are positive, then

$(\sin\theta+\cos\theta)/(\cos\theta(1-\cos\theta))=(391)/(90)$

• If both are negative, then

$(\sin\theta+\cos\theta)/(\cos\theta(1-\cos\theta))=(391)/(480)$

• If sin is positive and cos is negative, then

$(\sin\theta+\cos\theta)/(\cos\theta(1-\cos\theta))=(119)/(480)$

• If cos is positive and sin is negative, then

$(\sin\theta+\cos\theta)/(\cos\theta(1-\cos\theta))=(119)/(30)$