Hi Steve and Damon, if you're reading this, please take a look at my previous post because I had to apologize for my mistake.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ My Home Work Question:
Suppose that α is an angle such that tanα=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).
so, use the cos formula, instead of sin. What's the trouble?
cos(α−β) = cosα cosβ + sinα sinβ
You have all the values; just plug 'em in.
As Steve and Damon pointed out to you earlier,
sin(α−β) = sinαcosβ - cosαsinβ
given: tanα=5/7 and sinα<0 , so by the CAST rule
α must be in quadrant III
since tanα = 5/7, by Pythagoras, the hypotenuse of our triangle is √(25+49) = √74
thus sinα = -5/√74 and cosα = -7/√74
similarily, if cotβ=−3/5 and cscβ>0 , β must be in quadrant II, and tanβ = -5/3
sinβ = 5/√34 , and cosβ = -3/√34
You now have the exact 4 values needed to sub into my formula in the second line.
THANK YOU GUYS SOOO MUCH, GOD BLESS.
To find the exact value of sine(α−β), we will use the angle difference formula for sine:
sin(α-β) = sinα * cosβ - cosα * sinβ
First, let's find the values of sinα, cosα, sinβ, and cosβ by using the given information about α and β.
From the given information:
tanα = 5/7
Since tan is positive in the first and third quadrants, we know that α is in either the first or third quadrant. Also, we know that sinα < 0, which means α is in the third quadrant.
In the third quadrant:
sinα is negative (since sine is negative in the third quadrant), and we can use the Pythagorean identity to find cosα:
sinα = -sqrt(1 - cos^2α)
Solving for cosα:
sinα = -sqrt(1 - cos^2α)
(sinα)^2 = 1 - cos^2α
cos^2α = 1 - (sinα)^2
cos^2α = 1 - (25/49)
cos^2α = 49/49 - 25/49
cos^2α = 24/49
cosα = -sqrt(24/49) = -2sqrt(6)/7
Similarly, we can find the values of β:
cotβ = -3/5
Since cot is negative in the second and fourth quadrants, we know that β is in either the second or fourth quadrant. Also, we know that cscβ > 0, which means β is in the first or the second quadrant.
In the second quadrant:
cotβ = -3/5
cotβ = cosβ/sinβ
Since cotβ is negative and cos is negative in the second quadrant, and sin is positive in the second quadrant, we can consider cosβ as -3 and sinβ as 5, and scale it down if needed.
Let's find sinβ:
sinβ = √(1 - cos^2β)
sinβ = √(1 - 9/25)
sinβ = √(16/25)
sinβ = 4/5
Now, substitute the values of sinα, cosα, sinβ, and cosβ into the angle difference formula for sine:
sin(α-β) = sinα * cosβ - cosα * sinβ
sin(α-β) = (sinα * cosβ) - (cosα * sinβ)
sin(α-β) = (-sqrt(1 - cos^2α) * cosβ) - (-2sqrt(6)/7 * 4/5)
Now, simplify the expression to get the exact value of sine(α-β).