given that sin(a+b)= 56/65, cos(a+b)-33/65, cos(b)=-12/13 and pi/2<b,pi/
Find:
sin(a)?
cos(a)?
tan(a)?
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To find sin(a), cos(a), and tan(a), we'll use the given information and the trigonometric identities.
First, let's solve for a+b using the given values:
cos(b) = -12/13
cos(a+b) = 33/65
We know that cos(b) = adjacent/hypotenuse, so we can identify the adjacent side as -12 and the hypotenuse as 13. By using the Pythagorean identity, we can calculate the opposite side as:
opposite side = √(hypotenuse^2 - adjacent^2)
opposite side = √(13^2 - (-12)^2)
opposite side = √(169 - 144)
opposite side = √25
opposite side = 5
Now, we have enough information to find sin(a+b) using the given values of cos(a+b) and the Pythagorean identity:
sin(a+b) = opposite/hypotenuse
sin(a+b) = 5/13
Since sin(a+b) = 56/65, we can set up an equation using the sum-to-product identity:
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
56/65 = sin(a)(-12/13) + cos(a)(5/13)
Now, let's solve this equation for sin(a) and cos(a):
56/65 = -12sin(a)/13 + 5cos(a)/13
Multiply through by 13 to eliminate the denominators:
56 = -12sin(a) + 5cos(a)
To find sin(a), we'll rearrange the equation:
12sin(a) = 5cos(a) - 56
Square both sides:
(12sin(a))^2 = (5cos(a) - 56)^2
Using the trigonometric identity sin^2(a) + cos^2(a) = 1, we can substitute 1 - cos^2(a) for sin^2(a):
(12(1 - cos^2(a)))^2 = (5cos(a) - 56)^2
Expanding the equation:
144 + 288cos^2(a) - 144cos^4(a) = 25cos^2(a) - 560cos(a) + 784
Rearrange the equation:
144cos^4(a) + (25 - 288)cos^2(a) + 560cos(a) - 784 + 144 = 0
Simplify the equation:
144cos^4(a) - 263cos^2(a) + 560cos(a) - 640 = 0
Now, we can solve this equation to find the values of cos(a).
Once we have cos(a), we can use the trigonometric identity sin(a) = √(1 - cos^2(a)) to find sin(a) and tan(a) = sin(a)/cos(a) to find tan(a).
Unfortunately, the calculations for cos(a) involve solving a quartic equation, which does not have a simple closed-form solution. However, you can use numerical methods or software to approximate the values of cos(a), sin(a), and tan(a) based on the given values.