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.

Must show work!

1)  sin() =      (Decimal:  -0.258. . .)

2)  π/24, 5π/24, 13π/24, 17π/24

3) The replacement ramp forms an angle of 14.1 ° with the horizontal

Step-by-step explanation:

1)

sin()

use the following property: sin(-x) = -sin(x)

sin() = -sin()

sin() =

=

2)  

2sin(4x) = 1

sin(4x) = 1/2

4x = arcsin(1/2) = π/6, 5π/6, plus those +2π, 4π, etc

 π/6+2π = π/6+12π/6 = 13π/6

 5π/6+2π = 5π/6+12π/6 = 17π/6

 π/6+4π = π/6+24π/6 = 25π/6

 5π/6+4π = 5π/6+24π/6 = 29π/6

x = above divided by 4, so solutions are

(1/4)(π/6, 5π/6, 13π/6, 17π/6)

or

π/24, 5π/24, 13π/24, 17π/24

3)

To carry out this problem we have to think of the ramp as a right triangle. The length of the ramp will be the hypotenuse and the height will be the opposite leg to the angle they give us.

To calculate the degree of the new ramp we have to first calculate the height of our ramp that would be the same on the 2 ramps

well to start we have to know the relationship between angles, legs and the hypotenuse in a right triangle

α = 24°

a: adjacent  

o: opposite  = ?

h: hypotenuse  = 30

sin α = o/h

cos α = a/h

tan α = o/a

we see that it has (angle, hypotenuse, opposite)

is the sine sin α = o/h

sin 24° = o/30

sin 24° * 30 = o

12.2 = o

The height of the ramp is 12.2ft

Now we have the length of the hypotenuse and the opposite leg and we can calculate the angle

sin α = o/h

sin α = 12.2/50

α = sin^-1(12.2/50)

α = 14.12°

if we round to the nearest tenth

α = 14.12° = 14.1°

44

step-by-step explanation: