Thank you Problem 7 1Bonus Establish the identity 1 cos28 c

Problem 7 1Bonus] Establish the identity. 1 + cos(28)+ cos(40)+ cos(6e)- 4 cos(e) cos(28) cos (36) Problem 8 (Bonus] Show that sin(sin-(x)+cos(x))1 Problem 9 IBonus!.In a laser projection system, th distance D from the scanner to the screen and the projected image width W by the equation s] in a laser projection system, the optical angle or scanning angle is related to the throw a. Show that the projected image width is given by w = 2D tang b. Find the optical angle if the throw distance is 15 feet and the projected image width is 6.5 feet. Problem 10 Bonus] An object is propelled upward at an angle e,45\'s 90, to the horizontal with an initial velocity of vo feet per second from the base of a plane that makes an angle of 45* with the horizontal See the function illustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the R)-2Cos (0)(sin()-cos(o) 16 a. Show that R(0) = (sin(20)-cos(20)-1) 32 b. In calculus, you will be asked to find the angle 6 that maximize R by solving the equation sin(28) + cos(28)0 Solve this equation for .

Solution

problem 7:

1+cos(2)+cos(4)+cos(6)

=[1+cos(4)]+[cos(6)+cos(2)]

=[(cos²(2)+sin²(2))+(cos²(2)-sin²(2))]+[2cos((6+2)_/2)cos((6-2)_/2)]

=2cos²(2)+ 2cos(4)cos(2)

=2cos(2)[cos(2)+cos(4)]

=2cos(2)[cos(4)+cos(2)]

=2cos(2)[2cos((4+2)_/2)cos((4-2)_/2)]

=2cos(2)[2cos(3)cos()]

=4cos()cos(2)cos(3)

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problem 8:

sin(A+B)=sinAcosB +cosAsinB

sin(sin^-1(x)+cos^-1(x))

=(sin(sin^-1(x))cos(cos^-1(x))) +(cos(sin^-1(x))sin(cos^-1(x)))

=(x*x)+ (cos(cos^-1((1-x²))) *sin(sin^-1((1-x²))))

=(x²)+((1-x²)*(1-x²))

=(x²)+(1-x²)

=1