Sona 2 years ago how wud u solve the following? $\cos (\tan^{-1} (-5/12)+\tan^{-1} (15/8))$ Plz help this is in 15 min!

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1. cnknd

ok do the numbers (5,12, ) and (8,15, ) remind you of something?

2. cnknd

these numbers appear in the problem for good reason

3. Sona

umm im not sure way u mean

4. cnknd

ok let me fill in the blanks: (5,12,13), (8,15,17). do these numbers look familiar?

5. Sona

these r (x y r) values?

6. cnknd

pythagorean triples. i.e. integer side lengths for right triangles

7. Sona

ok then wat do i do w/this #?

8. Sona

don't mean to rush but this is due online in 10 min exactly

9. cnknd

so you know [\tan^{-1} (-5/12)] refers to the 5,12,13 triangle, and [\tan^{-1} (15/8)] refers to the 8,15,17 triangle

10. cnknd

ugh sry not familiar with the mathtype syntax here... i apologize for poorly formatted equations

11. Sona

its ok don't worry bout its understandable

12. cnknd

now the 2 inverse tangents give you to angles. let's call them a and b. the angle a comes from the 5,12,13 triangle, and the angle b comes from the 8,15,17 triangle

13. cnknd

draw your triangles, and identify which angle is a, and which angles is b

14. cnknd

now back at the original equation, since i'm lazy and don't want to write out all the inverse tangents, i'll just write a and b: cos(a+b) = cos(a) cos(b) - sin(a) sin(b)

15. cnknd

from your triangles, you should be able to identify what cos(a), cos(b), sin(a), and sin(b) are

16. Sona

so y for the 1st one is 5 n x is 12 while the 2nd y is 15 n 2nd x is 8?

17. cnknd

|dw:1353483877024:dw|

18. cnknd

sry, diagram not drawn to scale

19. Sona

thats fine but I'm still stuck :(

20. cnknd

ok back to original equation remember a = tan-1(5/12) and b = tan-1(15/8) so the original equation: cos(a+b), can be expanded as cos(a)cos(b) - sin(a)sin(b)

21. cnknd

now look at the pictures, what's cos(a)? what's cos(b)? what's sin(a)? what's sin(b)? (remember cos is adjacent/hypotenuse, sin is opposite/hypotenuse)

22. akinfermo

you should make a change of variables