Question 67322This question is from textbook Advanced mathematics
: Evaluate without a calculator:
A) sin{ Arctan [(-5/13)]}
B)cos{ Arcsin[(-3/4)]}
C)Arccos[(-square root of 3/2)]
D) tan { Arcsin[(-3/4)]}
This question is from textbook Advanced mathematics
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Evaluate without a calculator:
A) sin{ Arctan [(-5/13)]}
arctan(-5/13)is in the 4th quadrant by definition.
Sin is negative in the 4th quadrant.
Draw a rt. triangle with angle theta, side opposite theta = -5; side
adjacent to theta = 13.
Use Pythagoras to find the hypotenuse = 13.928
Therefore sin(arctan(-5/13))=-5/13.928
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B)cos{ Arcsin[(-3/4)]}
arcsin(-3/4) is in the 4th quadrant.
cos is positive in the 4th quadrant.
Draw a right triangle with angle theta,side opp theta = -3, hyp=4
Use Pythagoras to find side adjacent to theta = sqrt7
Therefore cos(arcsin(-3/4))=sqrt7/4=appoximately 0.6615
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C)Arccos[(-square root of 3/2)]
arccos(negative) is in the 2nd quadrant by definition.
If cos theta = (sqrt3)/2, theta=30 degrees
But the angle is in the 2nd quadrant so =180-30=150 degrees
Arccos((-sqrt3)/2)=150 degrees
D) tan { Arcsin[(-3/4)]}
arcsin(negative) is in the 4th quadrant
tan is negative in the 4th quadrant
Draw a triangle with angle theta, side opp theta=-3, adj hyp=4
Use Pythagoras to find adjacet = sqrt(4^3-3^2)=sqrt7
tan(arcsin(-3/4)=opp/adj=-3/sqrt7 = -3/7(sqrt7)
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Cheers,
Stan H.
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