Help please?

In the cast diagram how do wenter image source here e know whether we're supposed to go anticlockwise or clockwise? In part a) tanx is negative so WHY are we going anticlockwise (ie 180 - 66.8 and 360 - 66.8)? And in part b I dont get the cast diag. Could someone pls help me out with this?

2 Answers
Mar 27, 2018

For the first part:
tan will give the negative value at the 1st and fourth quadrant only

For question 2:
On the right hand side, you need to apply a trigonometry identity which is sin^2theta=1-cos^2theta
Then rearrange the equation so that all will be on the left hand side, which will give:
21cos^2theta+costheta-2=0

Let cos theta=x
therefore 21x^2+x-2=0

Factor the equation:
(7x-2)(3x+1)=0
7x-2=0 and 3x+1=0

x=2/7 and x=-1/3

Substitute back x=costheta
costheta=2/7 and costheta=-1/3

Then notice that theta give positive value on 1st and 4th quadrant while negative value on 2nd and 3rd quadrant

I hope that helps!

Mar 27, 2018

Please see below.

Explanation:

.

Whenever you solve for an angle from a trigonometric equation, you always have to go counterclockwise to get a positive angle and clockwise to get the a negative angle that has the same magnitude.

a) This is regardless of whether the angle function is negative or positive. There are usually four angles whose function values are the same, and this is how to find them.

b)

costheta=2/7

arccos(2/7)=1.28 radians

Again, based on the explanation in a), this is the positive angle, i,e, when you go counterclockwise. You also have to go clockwise to get the negative angle which is -1.28 radians.

If after reaching 1.28 radians, going counterclockwise, you continue to open up the angle and reach the terminal side of it when it falls on the terminal side of -1.28 radian angle, in reality, you have gone 2pi-1.28 radians which is equal to 2(3.14)-1.28=6.28-1.28=5 radians.

Similarly, if after reaching -1.28 radians, going clockwise, you continue to open up the angle until its terminal side falls on the terminal side of the 1.28 radian angle, in reality, you have gone -2pi+1.28=-6.28+1.28=-5 radians.

All four angles are in quadrants 1 and 4 where the cos is positive.

Similarly:

arccos(-1/3)=1.91 radians. Going clockwise, you get -1.91 radians. 1.91=pi-1.23

Following the same process as before, we go 1.23 radians past pi and get:

pi+1.23=4.37 radians. Both of these angles are positive and have the same cos value. Following the same process as before, going clockwise, we get two negative angles that are:

-pi+1.23=-3.14+1.23=-1.91 radians, and

-pi-1.23=-3.14-1.23=-4.37 radians.