Finding the intersection points on the graph y=sinx, y=cosx and y=tanx

[bersim]

Finding the intersection points

Hi guys, I'm new to this site and it seems like it will be a great resource when I'm stuck on a problem. I'll firstly set out the question and then add in my working so far.

Question: I was firstly asked to graph the functions y=sinx, y=cosx and y=tanx. My question is to find the points of intersection of where those lines cross. So far I have found the intersection points for where y=cosx and y=sinx cross (hopefully I'm right).

I'm now struggling with finding the intersection points for where y=cosx and y=tanx cross as well as the points for where y=sinx and y=tanx cross.

I know that the points for y=sinx and y=tanx will be (0,0) as it can easily be see from the graph. This is my working so far for that...

1. y=sinx
2. y=tanx

Sub 1. into 2.
sinx=tanx

Using the identity:
tanx=sinx/cosx

(I'm not too sure here if I'm meant to be finding an angle?)
sinθ=sinθ/cosθ
cosθ x sinθ = sinθ
cosθ=1
θ=cos^-1 (1)
θ=0

I'm not really sure where to go from here to show that it must have an intersection point of (0,0) - It would be greatly appreciated if someone could please help me on this.

Lastly, I'm also not sure how to find the intersection points for where y=cosx and y=tanx cross.

For this I really haven't got much at all. I do know that I must use the identity sin² x + cos² x = 1 (I'm pretty sure). At the moment this is all I have...

1. y=cosx
2. y=tanx

sub 1. into 2.
cosx=tanx
cosx=sinx/cosx
cosx x cosx = sinx
(This step below may be wrong?)
cos²x=sinx

Using the identity sin² x + cos² x = 1
cos² x=1-sin² x

From here I'm stuck on how to find the intersection points?

If anyone could help me find these intersection points for the 2 questions above I would be very grateful. Sorry about the lengthy thread.

Thanks in advance!
Jack

 

[pka]

Question: I was firstly asked to graph the trigonometric functions y=sinx, y=cosx and y=tanx in the interval where x is greater than or equal to 0 but less than or equal to pi/2. My question is to find algebraically the points of intersection of the three curves in that interval. So far I have found the intersection points for where y=cosx and y=sinx cross (hopefully I'm right - I got root 2/2 for both coordinates).

Shame on you: you should know the basic values by memory.
Look at the graphs. Clearly there is no common point of intersection.
Thus, the question must mean, find the points of pair-wise intersections.

From common knowledge:
\(\displaystyle \tan(x)~\&~\sin(x)\) intersect for \(\displaystyle x=0\) and \(\displaystyle \cos(x)~\&~\sin(x)\) intersect for \(\displaystyle x=\frac{\pi}{4}\).
Can you solve \(\displaystyle \sin^2(x)+\sin(x)-1=0~?\)

 

[Ishuda]

First, let's look at the answer to (1) sin(θ)=cos(θ): If cos(θ)=0 then sin(θ)\(\displaystyle \ne\)0 so we can assume cos(θ) is not zero and divide through by it to get
tan(θ) = 1
θ = tan^-1(1)
so your answer is incorrect in the value of θ. However, you were correct in the value of sin(θ) and cos(θ) where they intersect, i.e. \(\displaystyle \frac{\sqrt{2}}{2}\). But then maybe I misunderstood and that is what you meant. In doing work for others to judge, that is homework, etc., you need to be careful to be clear.

Next, let's look at (2) sin(θ) = tan(θ) at the point in your workings
sinθ=sinθ/cosθ
cosθ x sinθ = sinθ
cosθ=1
θ=cos^-1 (1)
θ=0
You are correct. However, you can divide through by sinθ to get θ=0 ONLY IF sinθ is not zero. This time you were lucky because, if sinθ=0, you would gotten the same point but that is not always the case.

Now for the third case (3) cos(θ) = tan(θ). You are correct up to the point
cos²(θ) = sin(θ)
but in using the identity sin²(θ) + cos²(θ) = 1, you substituted for the wrong thing. You should have substituted for the cos²(θ) to get
1 - sin²(θ) = sin(θ)
which becomes a quadratic equation is sin(θ) and thus has two values for sin(θ). However, only one will be in your interval of consideration.

 

[bersim]

Thanks for all the help guys, greatly appreciated!

Cheers, Jack