# Thread: Finding solutions within the range?

1. ## Finding solutions within the range?

Hi folks,new to the forum and would like to start by asking for help in four trignometry questions,lol!

As the title says, I have to find all solutions within the given range, Ive been racking my brains all day and can't get anywhere. They are as follows:

8.2 tan 2x = 10 0< x <180

5 cos 8 ( θ - 16) = -13 0 < θ < 140

5 sin 6x = -2 0< x < π

50 sin (3 θ + π/6) = 30 0< θ <2.5

π = pi (it doesnt appear to be clear the way ive written it

any help would be highly appreciated.

Thanks.

2. well you do the first one like this:

8.2 tan 2x = 10 0< x <180

divide the 8.2 to the other side
tan 2x = 10/8.2

take the inverse of tangent which makes it
tan^-1 (10/8.2) = 2x

divide the two so your left with x
and solve with a graphing calculator
[tan^-1 (10/8.2)]/ 2 = x = 25.324

then you can check 8.2*tan(2*25.324) = 9.999999 close enough haha
hope that atleast gets you on the right track

3. Originally Posted by Tino1210

50 sin (3 θ + π/6) = 30 0< θ <2.5

Thanks.
$50\sin(3\theta+\frac{\pi}{6})=30$

Divide both sides by 50.

$\sin(3\theta+\frac{\pi}{6})=\frac{3}{5}$

Take arcsin of both sides.

$\sin^{-1}(\sin(3\theta+\frac{\pi}{6}))=\sin^{-1}(\frac{3}{5})$

$3\theta+\frac{\pi}{6}=\sin^{-1}(\frac{3}{5})$

Subtract $\frac{\pi}{6}$ from both sides and divide by $3$

$\theta=\frac{1}{3}\sin^{-1}(\frac{3}{5})-\frac{\pi}{18}$

$\theta=.0399674444 \; \; radians$

Check: $50\sin(3(.0399674444)+\frac{\pi}{6})=30$