Two Curves Intersects

[kittenfelidae]

Hey there. Can someone please tell me how should I proceed for these types of questions. I have included the answers in red. Thanks

 

[Dr.Peterson]

First write the equation that says the two curves intersect: tan(x) = cos(x).

Then solve it.

What ideas do you have for doing this?

Hint: I'd probably start by expressing everything in terms of sine and cosine, though there are probably other ways that can work.

 

[MarkFL]

Hello, and welcome to FMH!

I would begin by equating the two curves, and solve for \(x\):

[MATH]\cos(x)=\tan(x)[/MATH]
Multiply by \(\cos(x)\) to obtain:

[MATH]\cos^2(x)=\sin(x)[/MATH]
Apply a Pythagorean identity to the LHS:

[MATH]1-\sin^2(x)=\sin(x)[/MATH]
At this point, we should recognize that we have a quadratic equation in \(\sin(x)\) and arrange in standard form:

[MATH]\sin^2(x)+\sin(x)-1=0[/MATH]
Can you proceed?

 

[kittenfelidae]

Hello, and welcome to FMH!

I would begin by equating the two curves, and solve for \(x\):

[MATH]\cos(x)=\tan(x)[/MATH]
Multiply by \(\cos(x)\) to obtain:

[MATH]\cos^2(x)=\sin(x)[/MATH]
Apply a Pythagorean identity to the LHS:

[MATH]1-\sin^2(x)=\sin(x)[/MATH]
At this point, we should recognize that we have a quadratic equation in \(\sin(x)\) and arrange in standard form:

[MATH]\sin^2(x)+\sin(x)-1=0[/MATH]
Can you proceed?

Thank you so much Mark. I think I'll manage from here.