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)?
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.
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 )
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
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.