Special triangles and Unit Circles HELP

1. Jan 9, 2005

aisha

If $$sin theta = \frac {-4} {7} and \frac {3 pi} {2} < theta < 2 pi$$then determine the exact value of $$\frac {1} {cot (theta)}$$

I dont know where to start I know to set up a circle with a cartesian plane but what am I supposed to do? :uhh:

Last edited: Jan 9, 2005
2. Jan 9, 2005

dextercioby

Okay,lemme get it straight...
You're given that
$$\sin\theta=-\frac{4}{7}$$ (1)
and
$$\frac{3\pi}{2}<\theta <2\pi$$ (2)
$$\frac{1}{\cot\theta}$$ (3)

Then the angle is in the IV-th quadrant where the 'sine' is negative and the cosine is positive.

Use the formula
$$\sin^{2}\theta+\cos^{2}\theta =1$$(4)
and the fact that 'cosine' is positive to find the 'cosine'.
Then
$$\frac{1}{\cot\theta}=\tan\theta=\frac{\sin\theta}{\cos\theta}$$(5)

,okay??

Daniel.

Last edited: Jan 9, 2005
3. Jan 9, 2005

aisha

Ok how do u know that the angle is in the 4th quadrant? How do u use that formula?? Im way too confused on this one y does it say <theta , 2 pi?

4. Jan 9, 2005

dextercioby

I guess,though u've been working with the trig.circle,u've never made a sign table for 'sine' and 'cosine'.
Besides,the 4-th quadrant is defined by
$$\frac{3\pi}{2}<\theta < 2\pi$$

Which formula to use???

Daniel.

5. Jan 9, 2005

aisha

ok yes I understand how the angle is in the forth quadrant, how do u use the eqn $$\sin^{2}\theta+\cos^{2}\theta =1$$
?? What do I have to do to solve this question are there any diagrams involved?

6. Jan 9, 2005

dextercioby

That is a second order algebraic eq.in 'cosine' of theta,since u know that the sine is -4/7.Solve it,knowing that u must accept the positive value.

Daniel.

PS.No diagrams,just algebra.

7. Jan 9, 2005

aisha

when u say solve what do u mean?? Does -4/7 mean that the opposite is -4 and 7 is the hypotenuse?

Im not sure how to solve for cos?

8. Jan 9, 2005

thursdaytbs

sin(theta) = -4/7, cos(theta) = unknown.

but you know:

sin^2(theta) + cos^2(theta) = 1
(-4/7)^2 + cos^2(theta) = 1

solve for Cos, and since it's a squareroot you're going to get a positive and negative answer, take the positive answer (since cos's positive in the restrictions given of 3pi/2 < theta < 2pi)

9. Jan 10, 2005

dextercioby

$$\sin^{2}\theta=(\sin\theta)^{2} =\frac{(-4)^{2}}{7^{2}}=\frac{16}{49}$$
$$\cos^{2}\theta=1-\sin^{2}\theta=1-\frac{16}{49} =\frac{33}{49}$$
$$\cos\theta=\pm\sqrt{\cos^{2}\theta}=\pm \frac{\sqrt{33}}{7}$$
U need to chose the plus sign (remember that the angle is in the IV-th quadrant,where the 'cos' is positive)
$$\cos\theta=\frac{\sqrt{33}}{7}$$

So now find the tangent.

Daniel.

10. Jan 10, 2005

thursdaytbs

OR you could draw a right triangle, make one angle theta, and put in 4 as the side opposite theta, and 7 as the hypotonose (since Sin = opposite/hypotonose = 4/7 [change -4 to 4 since you can't have a negative length of a side])

Use pythagorem theorm to find that the 3rd side = radical 33, and see that Cos(theta) = adjacent / hypotonose = radical 33 / 7.

*also keep in mind that cos must be positive since the restrictions given puts it in Q4 where cos is positive. so make cos positive and solve your equation knowing what sin= and what cos=.

Last edited: Jan 10, 2005
11. Jan 10, 2005

aisha

does $$\cos = sqrt (\frac {65} {49} )$$?

12. Jan 10, 2005

thursdaytbs

cos = (sqrt(33)) / 7

13. Jan 10, 2005

aisha

I dont know what Im doing wrong? The solution I have is $$\frac {-4} {sqrt 33}$$

14. Jan 10, 2005

dextercioby

That's the answer to the problem and it is correct.

Daniel.

15. Jan 10, 2005

aisha

lol but im still stuck :tongue2:

16. Jan 10, 2005

thursdaytbs

Yeah 1/cot = tan = -4 / sqrt33.

That's correct.

17. Jan 10, 2005

aisha

but how do u get that?

18. Jan 10, 2005

thursdaytbs

Tan = Sin / Cos.

You're given Sin = -4/7

Find Cos.

Sin^2 + Cos^2 = 1
(-4/7)^2 + Cos^2 = 1
Cos^2 = 1 - (-4/7)^2
Cos = SQUAREROOT ( 1 - (16/49))
Cos = Squareroot (33 / 49)
Cos = Squareroot(33) / 7
[You know squareroot of 49 doesn't = -7, and it = 7, because it's in Quadrent 4, where Cosine MUST be positive.]

So now you have: Sin = -4/7, Cos = Squareroot(33) / 7

1/Cot = Tan = Sin/Cos = (-4/7) / (squareroot(33)/7) = -4/squareroot(33)

Get it?

19. Jan 10, 2005

Zaimeen

aisha, it's best to draw a right angle triangle, with the hypotenuse being 7 and one of the sides being 4. Using Pythagoras theorem, the other side will be $$/sqrt 33$$

You know that cot theta = 1/(tan theta), right?

So, tan theta = (length_of_opposite) / (length_of_adjacent)

Hence, cot theta = (length_of_adjacent) / (length_of_opposite)

So, cot theta will be $$- \frac {/sqrt 33}{4}$$

20. Jan 10, 2005

HallsofIvy

With what? You were just told that the solution you give is correct.

The way I would approach this problem is:

First draw a picture: a unit circle on a pair of axes.

You are told that theta lies between 3pi/2 and 2 pi and you should know (MEMORIZE) that the axes correspond to 0 (at (1,0), pi/2 (at (0,1), pi (at (-1,0), 3pi/2 (at (0,-1) (and 2pi again at (1,0)). This is a circle with radius 1, diameter 2, and so circumference 2pi- those numbers just measure the distance around the circle.

Theta is between 3pi/2 and 2pi so you should mark a point on the circle between (0,-1) and (1,0) (i.e. in the fourth quadrant as dextercioby originally said). Since points on the unit circle have coordinates (cos(theta), sin(theta)), knowing that sin(theta)= -4/7 tells you that y= -4/7. The equation of the circle is x2+ y2= 1 so you must have x2+ 16/49= 1= 49/49 . That is:
x2= 33/49 and so x= cos(theta)= +&radic;(33)/7 (positive root because x is positive in the fourth quadrant).

Now that you know both sin(theta) and cos(theta) you know that 1/cos(theta)= tan(theta)= sin(theta)/cos(theta)= (-4/7)(7/sqrt(33))= -4/sqrt(33).

As for "where do I use $sin^2(x)+ cos^2(x)= 1$?", you don't HAVE to use it directly.. I basically used it when I wrote the equation of the circle, x2+ y2= 1, which, since x= cos(theta) and y= sin(theta), is the same thing.