Question

Help please?

In the cast diagram how do w 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?

Answer 1

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!

Answer 2

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.