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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
+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
+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
+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