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)
-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.
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
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.
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.
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.
To find the exact value of sine(α−β), we need to determine the values of sin(α) and sin(β) first. Given that tan(α) = 5/7 and sin(α) < 0, we can find sin(α) using the Pythagorean identity.
Step 1: Find cos(α)
The Pythagorean identity states that sin²(α) + cos²(α) = 1. Since tan(α) = sin(α)/cos(α), we can substitute tan(α) = 5/7 into the equation and solve for cos(α).
tan²(α) + 1 = cos²(α)
(5/7)² + 1 = cos²(α)
25/49 + 1 = cos²(α)
74/49 = cos²(α)
Taking the square root of both sides (keep in mind that cos(α) is positive since tan(α) is positive):
cos(α) = √(74/49)
cos(α) = √74/7
Step 2: Find sin(α)
Using the fact that sin²(α) + cos²(α) = 1, we can substitute the value of cos(α) we found in the previous step.
sin²(α) + (√74/7)² = 1
sin²(α) + 74/49 = 1
sin²(α) = 1 - 74/49
sin²(α) = 49/49 - 74/49
sin²(α) = (49 - 74)/49
sin²(α) = -25/49
Since sin(α) < 0, we take the negative square root:
sin(α) = -√(25/49)
sin(α) = -5/7
Now that we have sin(α), we can move on to finding sin(β). Given that cot(β) = -3/5 and csc(β) > 0, we can find sin(β) using the definitions of cotangent and cosecant.
Step 3: Find cos(β)
cot(β) = cos(β)/sin(β)
(-3/5) = cos(β)/sin(β)
Multiplying both sides by sin(β):
(-3/5) * sin(β) = cos(β)
Step 4: Find sin(β)
Using the Pythagorean identity, sin²(β) + cos²(β) = 1, we can substitute the expression for cos(β) we found in the previous step.
(sin(β))² + [(-3/5) * sin(β)]² = 1
sin²(β) + (9/25) * sin²(β) = 1
[1 + (9/25)] * sin²(β) = 1
(34/25) * sin²(β) = 1
Dividing both sides by (34/25):
sin²(β) = 25/34
Taking the square root of both sides (since csc(β) is positive):
sin(β) = √(25/34)
sin(β) = 5/√34 * √34/√34
sin(β) = 5√34/34
Lastly, we can now find the exact value of sine(α−β) using the angle subtraction formula: sin(α−β) = sinα⋅cosβ−cosα⋅sinβ.
Step 5: Find sin(α−β)
sin(α−β) = sinα⋅cosβ−cosα⋅sinβ
= (-5/7) * cos(β) - (-√74/7) * (5√34/34)
Simplifying:
= -5√34/7 * cos(β) + 5√74/14
= -5(√34/7)(-3/5) + 5√74/14
= 15√34/35 + 5√74/14
= (30√34 + 5√74)/70
Therefore, the exact value of sine(α−β) is (30√34 + 5√74)/70.