Question

Prove that tan(90+theta)= -cot Theta

Answer 1

welcome to openstudy @rayo

Answer 2

@50_cent Thank you! do you think you can help me?

Answer 3

u welcome and I will try as hard as I can

Answer 4

Thank you I appreciate it!

Answer 5

imagine a slope at an angle of say 30degrees(or theta) to the horizontal. you have a car on the slope moving downwards(not important). it has a weight perpendicular to the horizontal; and a contact force perpendicular to the slope, i.e. at 30degrees(or theta) to the horizontal. picture this.

now you want to break the weight into its 2 components: one will be perpendicular and the other will be parallel to the slope.

using simple mathematics the angle between the weight and the component of weight perpendicular to the slope(found below the slope) are the same. this component is the y component, while the other component, parallel to the slope, is the x component.

if you use the cosine of this angle, 30degrees(or theta), the closest component will be mg cos 30 (or mg cos theta). the component, farther, will be mg sin 30 (or mg sin theta).

now you can use the angle between the component parallel to the slope and the weight. this is equal to 60degrees (or 90 - theta).

if you use cosine of this angle, 60degrees (or 90 - theta), the closest component will be mg cos 60 (or mg cos (90 - theta)). the component, farther, will be mg sin 60 (or mg sin (90 - theta)).

you really need to visualize it to understand, i couldn't upload the diagram.

and i used weight as example. and this is a simple way to know which is sin and which is cos. the logic behind this is found by using trigonometrial reasoning i guess @rayo

Answer 6

and let me give u the answer

Answer 7

sin (90-theta) = cos theta
sin 90 - sin theta = cos theta

sin 90 =1

1 - sin theta = cos theta

in rule: cos theta + sin theta = 1
cos theta = 1 - sin theta

cos theta = cos theta

Answer 8

there u go have a nice day

Answer 9

@50_cent you said sin(90-theta) = cos theta
my question is tan(90 + theta)= -cot theta!

Answer 10

thats what I put in my QA sorry it's tan(90-theta)= cos theta

Answer 11

tan(90 + theta)

Answer 12

tan(90+theta)= - cot theta

Answer 13

tan(90 + theta)

Answer 14

yes sir!

Answer 15

how do I prove that tan(90+theta) = -cot theta

Answer 16

hold on

Answer 17

@satellite73

Answer 18

@satellite73 can ya help me please?

Answer 19

@Noura11

Answer 20

@JA1

Answer 21

We have :
\[\tan(\theta+90^\circ)=\frac{\sin(\theta+90^\circ)}{\cos(\theta+90^\circ)}=\frac{\cos\theta\sin90^\circ+\sin\theta\cos90^\circ}{\cos\theta\cos90^\circ-\sin\theta\sin90^\circ}\\~~~~=\frac{\cos\theta}{-\sin\theta}=-\cot\theta\]

Answer 22

@noura Thank you very much can you explain to me what happens in the 2nd last stage?\[\cos \theta \sin 90 + \sin \theta \cos 90 \div \cos \theta \cos 90 - \sin \theta \sin 90\]

Answer 23

Don't forget that :
\[\cos90^\circ=0\\\sin 90^\circ=1\]

Answer 24

yup so that would leave cos theta + sin theta div cos theta - sin theta

Answer 25

NO ! Look again !

Answer 26

Never mind, Just saw it

Answer 27

@noura Thank you, you are awesome :)

Answer 28

@rayo You are welcome !