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.

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- Nov 10th 2007, 06:48 AMfinalfantasyFind the angles
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.

- Nov 10th 2007, 08:07 AMSoroban
Hello, finalfantasy!

Quote:

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?*

- Nov 10th 2007, 08:17 AMfinalfantasy
Those are 2 separate questions ^^;

Sorry about that.

1. Cos Θ = 0.2157

2. Tan Θ =-0.7147 - Nov 10th 2007, 12:54 PMSoroban
Hello, finalfantasy!

Don't you have inverse trig functions on your calculator?

Quote:

$\displaystyle 1.\;\;\cos\theta \:=\:0.2157 $

$\displaystyle \theta \:=\:\cos^{-1}(0.2157) \;\approx\;1.35,\;4.93$ radians

Quote:

$\displaystyle 2.\;\;\tan\theta \:=\:0.7147$

$\displaystyle \theta \:=\:\tan^{-1}(0.7147) \:\approx\:0.62,\;3.76$ radians

- Nov 10th 2007, 07:31 PMfinalfantasy
Hmm ... those were the answers on my sheet, I got the first answer for each of those questions, but how do you get the second? o-o;

- Nov 11th 2007, 03:37 AMMacleef
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