Competition - post a photo you've taken of nature >>

Trigonometric functions question

The angle q is greater than 90◦ and less than 360◦ and cosq = 2/3.
Find the exact value of tanq.

q=theta (not that it matters)

I'm basically stuck on that and have no idea how to actually work it out.

I know the answer is meant to be: minus root 5 over 2
but have no idea how to work it out...

Could anybody take me through it?
I'd like the sinq/cosq=tanq version of working it please, because i don't know how to work anything out that other way. Thanks.

Reply 1

electriic_ink

Use the Pythagorean trig identity and the graphs of sin x and cos x to determine sin q.

(edited 11 years ago)

Reply 2

user1-4

Original post by electriic_ink

Use the Pythagorean trig identity and the graph of sin x to determine sin q.

i've tried.
I keep getting root 5 over 2 but without the minus in front of it as my answer.
I don't understand where the minus comes from.

and is the pytharorean identity the sin^2q+cos^2q=1?
If so that's what i've been doing so far, but with no luck.

Reply 3

TenOfThem

Original post by user1-4

Could anybody take me through it?
I'd like the sinq/cosq=tanq version of working it please, because i don't know how to work anything out that other way. Thanks.

That is not a method to use for this ... if it was then, presumably you could do it

:confused:

So ... the method

sketch a right angled triangle

label an angle q

put the adjacent and hypotenuse measurements in using your given fraction

use pythagoras to work out the 3rd side

remeber that the square root can be + or -

use tan = opp/adj

think about the size of the angle and decide on +ve -ve values using CAST/graphs etc

Reply 4

TenOfThem

Original post by user1-4

I keep getting root 5 over 2 but without the minus in front of it as my answer.
I don't understand where the minus comes from.

Do you use CAST or Graphs to find angles?

Reply 5

electriic_ink

Original post by user1-4

I added that you need the graph of cos x too, since you do.

We know cos q > 0 and 90 < q < 360. Combined these give us that 270 < q < 360 (look at the graph of cos x). Looking at the graph of sin x, we see that since 270 < q < 360, sin q < 0. And so tan q < 0.

And yes, that is the Pyathg identity.

(edited 11 years ago)

Reply 6

user1-4

Original post by TenOfThem

Do you use CAST or Graphs to find angles?

um i don't what cast is... I don't use graphs. I always just use one identity or the other.

Original post by electriic_ink

alright i'll try and do this... How do looking at those graphs give me an answer though. Don't they only give me an approximation?

Reply 7

TenOfThem

Original post by user1-4

um i don't what cast is... I don't use graphs. I always just use one identity or the other.

Well that explains why you cannot find the other values

Reply 8

electriic_ink

Original post by user1-4

alright i'll try and do this... How do looking at those graphs give me an answer though. Don't they only give me an approximation?

No. The only think you know about sin q from the Pythagorean identity is that sin^2 q = 5/4 . This gives to possibilities: (a) sin q = sqrt(5)/2 and (b) sin q = -sqrt(5)/2 . You use the graphs and the range of q given to work out which one it is. (In this case we can show that sin q < 0 so (b) is the correct value).

(edited 11 years ago)

Reply 9

user1-4

alright well thanks guys. I guess I have enough info to keep reading over now until i get it.

Reply 10

F1Addict

Using the CAST diagram, you're told in the question the angle is greater than 90 and less than 360, so you're only concerned with the 2nd,3rd and 4th quadrants (i.e. S,T and C). Then the question also says that cos(x) = 2/3, which is positive, so you now are only concerned with the 4th quadrant, C. This is also the quadrant where tan is negative, which is where the minus comes from.

(if you don't know what a CAST diagram is:

The above diagram is generally referred to as a CAST diagram.)

The graphing method works also and is really useful to know, but the CAST diagram is generally more intuitive for most people.

Reply 11

TenOfThem

Angles that are between 0-90 and 270-360 have +ve cos (like you have)

For +ve tan you need to be 0-90 or 180-270 (else it is -ve)

So ... you know your angle is between 270-360 and therefor tan is -ve

So you use the -ve root

Reply 12

user1-4

Ok I think I understand the CAST thing. Last question is

if 0-90 positive for all
then is e.g. sine +ve for for 90-180 in addition to this and -ve everywhere else.

If yes, then although I have no idea how any of this works, i understand how to use it so thanks! :biggrin: