Question 160870This question is from textbook Basic Technical Mathematics
: Please help me :)
Find theta for 0 degrees less tha or equal to (
a) tan theta= 0.932, sin theta<0
b) sin theta= -0.192, tan theta<0
c) sec theta= 2.047, cot theta <0
d) cot theta=-0.3256,csc theta>0
#50.page 243
This question is from textbook Basic Technical Mathematics
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! a. tan theta = .932, sin theta < 0.
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in order for the tangent to be positive, the angle has to be in the first or third quadrant.
in order for the sin to be negative, the angle has to be in the third or fourth quadrant.
looks like the angle has to be in the third quadrant to satisfy both requirements.
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calculator gives theta = 42.984... degrees.
adding 180 to theta puts it in the third quadrant.
180 + theta = 222.984... degrees
tan 222.9842... = .932
sine 222.9842... = minus
the answer is then 180 + theta = 180 + 42.98421048 = 222.98421048.
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b. sin theta = -.192, tan theta < 0
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in order for the sine to be negative, the angle must be in the third or fourth quadrant.
in order for the tangent to be negative, the angle must be in the second or fourth quadrant.
looks like the angle has to be in the fourth quadrant to satisfy both sin and tangent requirements.
calculator gives theta as -11.069... degrees.
that's in the 4th quadrant but it's not showing the angle the way we want it.
to get it the way we want to show it, we need to add 360 degrees to it. -11.069... degrees + 360 degrees = 348.930... degrees.
sine of 348.930... = -.192
tangent of 348.930... is negative.
both requirements are satisfied.
angle is 348.930...
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c. secant theta = 2.047, cot theta < 0
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secant is 1/cosine.
in order for secant to be positive, the angle must be in the first or fourth quadrant
cotangent is 1/cotangent.
in order for cotangent to be negative, the angle must be in second or fourth quadrant.
looks like the angle must be in the fourth quadrant to satisfy both secant and cotangent requirements.
since secant is 1/cos, then cos must be 1/secant.
since calculator only solves for cos, we work with cosine.
secant is 2.047
1/secant equals cosine = .4885...
angle shows as 60.756...
to put it in the fourth quadrant subtract it from 360.
360 - 60.756... = 299.243...
secant of 299.243... equals 1/cosine equals 2.047 which is accurate.
tangent of 299.243... is negative, therefore cotangent is also negative.
answer is 299.243 degrees.
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d. cot theta = -0.3256, csc theta > 0
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for the cotangent to be negative, the angle must be in the second or fourth quadrant.
for the csc to be positive, the sin is also positive and the angle must be in the first or second quadrant.
this makes the angle must be in the second quadrant to satisfy both.
since cotangent is 1/tangent, we divide 1 by cotangent to get tangent.
then we can use the calculator.
calculator give angle as -71.964 degrees which puts it in the fourth quadrant.
to put it in the second quadrant, add 180 to it.
-71.964 + 180 = 108.035...
to prove this is the right angle, get the cotangent of it.
get the tangent and then take 1/tangent to get the cotangent.
that calculates to be -.3256 proving the angle is correct.
to get the cosecant, get the sine and then take 1/sine to get the cosecant.
since the sine is positive, the cosecant is also positive.
answer is 108.035... degrees.
cotantent is -.3256 and cosecant is positive.
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