Step 1: Evaluate each trigonometric function.
We need to find the values of sin(32π), tan(35π), cot(3−4π), and cos(65π).
For sin(32π):
The angle 32π is in the second quadrant.
sin(32π)=sin(π−3π)=sin(3π)=23
For tan(35π):
The angle 35π is in the fourth quadrant.
tan(35π)=tan(2π−3π)=−tan(3π)=−3
For cot(3−4π):
First, use the identity cot(−x)=−cot(x):
cot(3−4π)=−cot(34π)
The angle 34π is in the third quadrant.
cot(34π)=cot(π+3π)=cot(3π)=31
So,
cot(3−4π)=−31
For cos(65π):
The angle 65π is in the second quadrant.
cos(65π)=cos(π−6π)=−cos(6π)=−23
Step 2: Substitute the values into the expression for A.
A=sin(32π)tan(35π)cot(3−4π)cos(65π)A=(23)⋅(−3)⋅(−31)⋅(−23)
Step 3: Multiply the terms to simplify.
A=23⋅(−13)⋅(−31)⋅(−23)
Multiply the numerators:
3⋅(−3)⋅(−1)⋅(−3)=(−3)⋅(−1)⋅(−3)=3⋅(−3)=−33
Multiply the denominators:
2⋅1⋅3⋅2=43
Combine the numerator and denominator:
A=43−33
Cancel out 3 from the numerator and denominator:
A=−43
The simplified value is:
−43
That's 2 down. 3 left today — send the next one.
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This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Step 1: Evaluate each trigonometric function. We need to find the values of ((2)/(3)), ((5)/(3)), ((-4)/(3)), and ((5)/(6)). For ((2)/(3)): The angle (2)/(3) is in the second quadrant. ((2)/(3)) = ( - ()/(3)) = (()/(3)) = sqrt(3)2 For ((5)/(3)): The angle (5)/(3) is in the fourth quadrant. ((5)/(3)) = (2 - ()/(3)) = -(()/(3)) = -sqrt(3) For ((-4)/(3)): First, use the identity (-x) = -(x): ((-4)/(3)) = -((4)/(3)) The angle (4)/(3) is in the third quadrant. ((4)/(3)) = ( + ()/(3)) = (()/(3)) = (1)/(sqrt(3)) So, ((-4)/(3)) = -(1)/(sqrt(3)) For ((5)/(6)): The angle (5)/(6) is in the second quadrant. ((5)/(6)) = ( - ()/(6)) = -(()/(6)) = -sqrt(3)2 Step 2: Substitute the values into the expression for A. A = ((2)/(3)) ((5)/(3)) ((-4)/(3)) ((5)/(6)) A = (sqrt(3)2) · (-sqrt(3)) · (-(1)/(sqrt(3))) · (-sqrt(3)2) Step 3: Multiply the terms to simplify. A = sqrt(3)2 · (-sqrt(3)1) · (-(1)/(sqrt(3))) · (-sqrt(3)2) Multiply the numerators: sqrt(3) · (-sqrt(3)) · (-1) · (-sqrt(3)) = (-3) · (-1) · (-sqrt(3)) = 3 · (-sqrt(3)) = -3sqrt(3) Multiply the denominators: 2 · 1 · sqrt(3) · 2 = 4sqrt(3) Combine the numerator and denominator: A = -3sqrt(3)4sqrt(3) Cancel out sqrt(3) from the numerator and denominator: A = -(3)/(4) The simplified value is: -(3)/(4) That's 2 down. 3 left today — send the next one.
