Need some help trying to solve this problem

[abel muroi]

Determine for what values of t , sin t = cos t



A. t = pi/4 + pi * k, where k is an integer

B. no matter what value of t is, sin t is always different than cos t.

C. t = pi

D. t = 0

E. t = tan t



I think the answer is B, since I believe that Sin and Cos will always be different.

Can someone help me solve this problem?

 

[ksdhart]

Let's think about what you know about how to tackle problems like these. If you'd been given two other functions of t, let's say f(t)=t^2+3t and g(t)=t^3, and asked to find where those intersect, what would you do? Well, my first step would be to graph them to get a visual sense of the question. What if we did that for this problem? Graph sin(t) and cos(t). Do they ever intersect? If they don't, then you know that your initial guess of answer B was correct. If not, then continue on...

In the case of the earlier simpler example, f(t) and g(t) do intersect. You've seen the one point of intersection on the graph, so how would you find that point? Well, you'd set them equal and solve. Now let's try doing that for the sine and cosine functions. Set sin(t)=cos(t) to get an equation. How would you go about solving that equation? Think about what it means for two functions to be equal, and remember your trig identities.

 

[abel muroi]

Let's think about what you know about how to tackle problems like these. If you'd been given two other functions of t, let's say f(t)=t^2+3t and g(t)=t^3, and asked to find where those intersect, what would you do? Well, my first step would be to graph them to get a visual sense of the question. What if we did that for this problem? Graph sin(t) and cos(t). Do they ever intersect? If they don't, then you know that your initial guess of answer B was correct. If not, then continue on...

In the case of the earlier simpler example, f(t) and g(t) do intersect. You've seen the one point of intersection on the graph, so how would you find that point? Well, you'd set them equal and solve. Now let's try doing that for the sine and cosine functions. Set sin(t)=cos(t) to get an equation. How would you go about solving that equation? Think about what it means for two functions to be equal, and remember your trig identities.

I graphed sin(t) and cos(t) and they do intersect. So my first guess was incorrect.

I've set sin(t) and cos(t) to an equation. sin(t) = cos(t)... but i don't know what to do next..

 

[ksdhart]

Well, that's why I asked you really consider what it means for two functions to be equal. If you had two variables that were equal, say x = y, then what could you say about x/y? Given then that your two functions are sin(t) and cos(t), recall that tan(t)=sin(t)/cos(t). If you know tan(t) equals something, how would you go about solving for t?

 

[abel muroi]

Well, that's why I asked you really consider what it means for two functions to be equal. If you had two variables that were equal, say x = y, then what could you say about x/y? Given then that your two functions are sin(t) and cos(t), recall that tan(t)=sin(t)/cos(t). If you know tan(t) equals something, how would you go about solving for t?

Hmm I think I'm starting to get this now.

So since Sin(t) = Cos(t), and Tan(t) = Sin(t)/Cos(t)

So that means that the answer is t = tan(t)

 

[stapel]

So since Sin(t) = Cos(t), and Tan(t) = Sin(t)/Cos(t)

So that means that the answer is t = tan(t)

How did you get from the first line quoted above to the second? You started with what you'd been given:

. . . . .\(\displaystyle \mbox{Solve for }t:\, \sin(t)\, =\, \cos(t)\)

You took the next step that you were given, which was to divide through by cosine; then you simplified each side:

. . . . .\(\displaystyle \sin(t)\, =\, \cos(t)\)

. . . . .\(\displaystyle \dfrac{\sin(t)}{\cos(t)}\, =\, \dfrac{\cos(t)}{\cos(t)}\)

. . . . .\(\displaystyle \tan(t)\, =\, 1\)

How did you then get to t equalling 1?

Please show all of your steps. Thank you!