Suppose that α is an angle such that tan α=(5/7) and sinα<0. ​Also, suppose that β is an angle such that cotβ=−3/5 and cscβ>0. Find the exact value of sine(α−β)

Question

-Direction (Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

Please show work + answer, step-by-step so I can learn. Thanks.

Answer 1

since sin α < 0 α is in quadrant 3 or 4
since tan α is +, that puts α in quadrant 3
-5,-7, sqrt 74 triangle

since csc β > 0, β is in quadrant 1 or 2
since cot β is -, β is in quadrant 2
-3, 5, sqrt 34 triangle

now
sin (a-b) = sin a cos b - cos a sin b

= -5/sqrt74*-3/sqrt34 +7/sqrt74*5/sqrt34

etc

Answer 2

Hi Damon,

Thanks for finding the time of day to help me. However, I'm not as smart as you are and I'm sure you'll expect me to know the answer. I've tried putting in -5/sqrt74*-3/sqrt34 +7/sqrt74*5/sqrt34 in my calculator, but instead it comes out with a really ugly number and I'm sure it's the wrong answer. So, if you will be so kind to help me out here will be nice. Thank you. I have few more of questions like this to solve and I'm not even close to finding this answer.

Answer 3

forget the calculator.

-5/√74*-3/√34 + 7/√74*5/√34
= 15/√2516 + 35/√2516
= 20/√2516
= 5/√629

when doing stuff like this, it's better to leave the radicals in place. The decimal value is only an approximation. The radicals are exact.

Answer 4

Hi Steve,

Please accept my apologies. My initial question is wrong. They want me to find cos(α−β), not sine(α−β). I really am sorry. I had the same answer as 5/radical629, but when I typed it in my h/w answer box, it kept saying the answer was wrong. So I was wondering why. Then, I re-read the question. Sorry once again, it was a stupid mistake. I re-read my question after typing it, but I didn't see my error. So please help me find cos(α−β) of the question. THANK YOU SOO MUCH.