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CHAPTER 5 Practical Applications
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A 150-lb shell is dragged up a runway inclined 40 to the horizontal. Find (a) the force of the shell against the runway and (b) the force required to drag the shell. Ans. (a) 115 lb, (b) 96 lb
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Two forces, the first one of 50 lb and the second one of 120 lb, act on an object at right angles to each other. Find the magnitude of the resultant force and the measure of the angle the resultant force makes with the 120-lb force. Ans. 130 pounds, angle of 22.6 with the 120-lb force
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An airplane is flying on a heading of 45 with an airspeed of 500 mi/h and the wind is blowing on a heading of 315 at a speed of 75 mi/h. Find the drift angle, the groundspeed, and the course of the plane. Ans. Drift angle 8.5 , groundspeed 506 mi/h, and course 36.5
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The resultant of two forces acting at right angles to each other is 75 N. If one of the forces is 45 N, what is the other force and the angle it makes with the resultant Ans. 60 N, 37
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A steel ball, weighing 50 kg, is on an inclined surface. What angle does the inclined surface make with the horizontal if a force of 10 kg is exerted on the inclined surface Ans. 12
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Reduction to Functions of Positive Acute Angles
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6.1 Coterminal Angles
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Let be any angle; then sin ( cos ( tan ( n 360 ) n 360 ) n 360 ) sin cos tan cot ( sec ( csc ( n360 ) n360 ) n360 ) cot sec csc
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where n is any integer (positive, negative, or zero).
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EXAMPLE 6.1 (a) sin 400
sin (40 cos (130
360 )
sin 40 cos 130 tan 80 3 360 )
(b) cos 850
2 360 )
(c) tan ( 1000 )
tan (80
If x is an angle in radian measure, then sin (x cos (x tan (x
where n is any integer. EXAMPLE 6.2 (a) sin 11 /5 (c) tan 137 sin( /5 tan [ 2 ) 68(2 )] sin /5 2(2 )] tan cos 17 /11
2n ) 2n ) 2n )
sin x cos x tan x
cot (x sec (x csc (x
2n ) 2n ) 2n )
cot x sec x csc x
(b) cos ( 27 /11)
cos [17 /11
6.2 Functions of a Negative Angle
Let be any angle; then sin ( cos ( tan (
EXAMPLE 6.3 sin ( 50 ) see Prob. 6.1.
) ) )
sin cos tan
cot ( sec ( csc (
) ) )
cot sec csc
tan 200 . For a proof of these relations,
sin 50 , cos ( 30 )
cos 30 , tan ( 200 )
CHAPTER 6 Reduction to Functions of Acute Angles
6.3 Reference Angles
A reference angle R for an angle u in standard position is the positive acute angle between the x axis and the terminal side of angle u. The values of the six trigonometric functions of the reference angle for u, R, agree with the function values for u except possibly in sign. When the signs of the functions of R are determined by the quadrant of angle u, as in Sec. 2.5, then any function of u can be expressed as a function of the acute angle R. Thus, our tables can be used to find the value of a trigonometric function of any angle.
Quadrant for I II III IV
Relationship R R R R 360 180 180
Function Signs All functions are positive. Only sin R and csc R are positive. Only tan R and cot R are positive. Only cos R and sec R are positive.
See Prob. 6.2 for a verification of the equality of the values of the trigonometric functions of and signed values of its reference angle R. If is a quadrantal angle, then the function values are the same as in Sec. 2.6, and a reference angle is not needed. Since any angle A can be written as n360 , where n is an integer and 0 < 360 , reference angles will be found for angles from 0 to 360 .
EXAMPLE 6.4 Express each as a function of an acute angle. (a) sin 232 ,
(b) cos 312 ,
(c) tan 912 ,
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