Last 1 for finding exact values?

Question

Given that tan a = - 4/3, a lies in the quadrant II and cos b =2/3, b lies in quadrant I.
Find the exact value of tan(a+b)

Thanks for all the help. I have to wait 4 hours before choosing the best answer.

I don't get it. Can you draw the graph with MSPaint and upload it?

I don't get it. Can you draw the graph with MSPaint and upload it?
I know
cos(a) = -3/5
sin(a) = 4/5
cos(b) = 2/3
sin(b) = sqrt(5)/3

Okay, I have how you got tan(b). What do we do for tan(a)?

Okay, I got it!

Answer 1

You need the value of tan b and the formula
tan (a + b) = (tan a + tan b)/(1-tana tanb)

Draw a rightangled triangle with adjacent side 2, hypotenuse 3, then use Pythag to get the side opposite to angle b. (Hope you get sqrt(5))
Thus tan b = sqrt(5)/2, and since it's in quadrant 1 you know the positive sign is correct.

Sub in the formula for tan (a+b) and you have the answer.

To simplify it, multiply both top and bottom by 6, then to express with rational denominator I think you'll need to multiply top and bottom by (4*sqrt(5) - 6)

SORRY, I'M HOPELESS WITH PAINT AND WOULDN'T KNOW HOW TO UPLOAD IT ANYWAY! But you've already done it anyway. You know cos b = 2/3 and sin b = sqrt(5)/3,
and so since
tan b = (sin b)/(cos b), you just put one over the other, then multiply both top and bottom by 3 and you're left with
sqrt(5)/2 OK?

If not, email [e-mail], and I can email more detail. I can do graphics in WORD














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If you multiply both top and bottom by 6 you should get

(3*sqrt(5) - 8)/(4*sqrt(5) + 6) unless I've made a mistake.

Then, with rational denominator, I think that's

(54 - 25*sqrt(5))/22

Answer 2

Note that you're given these:

cos(a) = -3/5
sin(a) = 4/5
cos(b) = 2/3
sin(b) = sqrt(5)/3

From here, you can derive what tan(a) and tan(b) are.

Note that tan(x) = sin(x)/cos(x). Therefore

tan(a) = sin(a) / cos(a) = (4/5) / (-3/5) = -4/3
tan(b) = sin(b) / cos(b) = (sqrt(5)/3) / (2/3) = sqrt(5)/2

There's a formula for tan(a + b), and it goes

tan(a + b) = [tan(a) + tan(b)] / [1 - tan(a)tan(b)]

Plugging in your values,

tan(a + b) = [(-4/3) + (sqrt(5)/2)]/[1 - (-4/3)(sqrt(5)/2)]

Alternatively, if you know sin(a + b) and cos(a + b), you can divide them, since
tan(a + b) = sin(a + b) / cos(a + b).