Find the angle Θ, 0 ≤ Θ ≤ 2 pi
Cos Θ = 0.2157 Tan Θ =-0.7147
The answers I were given show 2 answers can be found. Can someone tell me the answers for both questions and explain how it can be found.
THANK YOU.
Hello, finalfantasy!
Find the angle $\displaystyle \theta,\;\;0 \leq \theta \leq 2\pi$
. . $\displaystyle \cos\theta = 0.2157,\;\;\tan\theta = -0.7147$
The answers show two answers can be found. . ??
Only one angle is possible.
Cosine is positive in Quadrants 1 and 4.
Tangent is negative in Quadrants 2 and 4.
. . Hence, $\displaystyle \theta$ is in Quadrant 4 . . . so where's the other angle?
Hello, finalfantasy!
Don't you have inverse trig functions on your calculator?
$\displaystyle 1.\;\;\cos\theta \:=\:0.2157 $
$\displaystyle \theta \:=\:\cos^{-1}(0.2157) \;\approx\;1.35,\;4.93$ radians
$\displaystyle 2.\;\;\tan\theta \:=\:0.7147$
$\displaystyle \theta \:=\:\tan^{-1}(0.7147) \:\approx\:0.62,\;3.76$ radians
Use the inverse function in your calculator,
1. Cos^{-1}(0.2157) = 78
2. Tan^{-1}(-0.7147) = -36
I think you want the answer in both approximate and exact form, right? Or, find the two angles?
To find the radians, you must convert degrees to radians, to do so:
(degree) x (pi / degree)
If you want to find the degrees from radians:
(radian) x (degree / pi)
1. Cos^{-1}(0.2157) = 78
78 x pi/180
= 78pi/180
=13pi/30 or 1.36 rad
To find the other angle in quadrant 4, you do your principle angle:
360 - Θ
= 360 - 78
= 282
282 x pi/180
= 282pi/180
= 14pi/9 or 4.89 rad
2. Tan^{-1}(-0.7147) = -36
Since the degree is a negative, the angle will be in the second and fourth quadrant because in the first and third quadrant, the angle would be in positive, hence, CAST rule
First, you must find your related angle, which is 36 in this case
36 x pi/180
= 36pi/180
= pi/5 or 0.63 rad
To find the other radian, again, you must do your principle angle
360 - Θ
= 360 - 36
= 324
324 x pi/180
= 9pi/5 or 5.65 rad