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1. What is the exact value of sin( -   pi/ 12)?

 

 

2. What are the solutions of the equation 2sin (4x)=1 on the interval [0,pi)?

 

X=____________

 

X=____________

 

X=____________

 

X=____________

 

 

3. A 30-foot ramp is used to move product to a loading dock. The ramp makes a 24° angle with the horizontal. The ramp is to be replaced with a ramp that is 50 feet long. What angle does this replacement ramp make with the horizontal? Round your answer to the nearest tenth of a degree. 

 Apr 19, 2020
 #1
avatar+20810 
+1

1)  Use the half-angle formula for sin:  sin(A/2)  =  +/- sqrt( (1 - cos(A) ) / 2 )

      where  A  =  pi/6     and choose the negative sign because the angle is in Quadrant IV.

      sin( - pi/12 )  =  - sqrt( (1 - cos( pi/6 ) ) / 2 )  =  -sqrt( 1 - sqrt(3)/2 ) / 2 )

 Apr 19, 2020
 #2
avatar+20810 
+1

2)  2sin(4x)  =  1     --->     sin(4x)  =  1/2     --->     4x  =  sin-1(1/2)     --->      x  =  sin-1(1/2) / 4

                               --->     x  =  ( pi/6 ) / 4     --->     x  =  pi/24

 

     So, the answers will be:  pi/24

                                             pi/4 - pi/24

                                             pi/24 + pi/2

                                             pi/4 - pi/24 + pi/2

 Apr 19, 2020
 #3
avatar+20810 
+1

3)  The triangle for the original ramp has an hypotenuse of 30' and an angle of 24o

      We need to find the height at the opposite end of the ramp.

      sin(24o)  =  height / 30     --->     height  =  30 · sin(24o)  =  12.2021'

 

      When we replace the ramp, this height will remain the same.

      Since the new ramp is 50' long, the hypotenuse of this triangle is 50'.

      sin(theta)  =  12.2021' / 50'     --->     theta  =  sin-1( 12.2021 / 50 )     --->     theta  =  14.1o.

 Apr 19, 2020

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