Trig verify-simplify
Prove sin(8x) = 2sin(4x)cos(4x)
Start with left side:
sin(8x)
sin(2*4x)
2sin(4x)cos(4x)
sin(390+45)
sin(30)cos(45)+cos(30)sin(45)
1/2 * sqrt2/2 + sqrt3/2* sqrt2/2
sqrt2/2(1/2+sqrt3/2)
sqrt2/2((1+sqrt3)/2)
sqrt2 + sqrt6/2
Using Diagram 1 on the paper provided, solve for x.
5
The previous official name of Watertown
What is the city known as the town of Watertown?
Prove (2sin(x)cos(x))/((cos(x)-sin(x))*(cos(x)+sin(x))) = tan(2x)
Start with left side:
(2sin(x)cos(x))/((cos(x)-sin(x))*(cos(x)+sin(x)))
(2sin(x)cos(x)) / (cos2(x)-sin2(x))
sin(2x)/cos(2x)
tan(2x)
1-(2-2cos2(pi/6))
1-(2-2cos2(pi/6))
1-2(1-cos2(pi/6))
1-2sin2(pi/3)
cos(2pi/6)
cos(pi/3)=1/2
Using Diagram 2 on the paper provided, solve for x (round to the nearest whole number).
7
The city manager of Watertown
Who is George Proakis?
Prove (sin(6x) + sin(8x)) / (cos(6x)-cos(8x)) = cot(x)
Start with left side:
(sin(6x) + sin(8x)) / (cos(6x)-cos(8x))
(2sin((6x+8x)/x)*cos((6x-8x)/2)) / (-2sin((6x+8x)/2)*sin((6x-8x)/x))
(2sin(14x/2))*cos(-2x/2)) / (-2sin(14x/2)*sin(-2x/2))
(2sin(7x)cos(-x)) / (-2sin(7x)sin(-x))
cos(-x)/-sin(-x)
-cot(-x)
-(-cot(x))
cot(x)
1-sin2pi-sin2pi + sin(30+45)
1-sin2(pi)-sin2(pi) + sin(30+45)
1-sin2(pi)-sin2(pi) +(sqrt2 + sqrt6)/2
cos2(pi)-sin2(pi) + (sqrt2 + sqrt6)/2
cos(2pi) + (sqrt2 + sqrt6)/2
1+(sqrt2 + sqrt6)/2
Using Diagram 3 on the paper provided, solve for x (round to the nearest whole number).
8
The number of parks in Watertown
What is 13?
Prove (2cos2(2x) - sin2(x)-cos2(x)) / (cos(3x)cos(x) - sin(3x)sin(x)) = 1
Start with left side:
(2cos2(2x) - sin2(x)-cos2(x)) / (cos(3x)cos(x) - sin(3x)sin(x))
(2cos2(2x) - 1) / (cos(3x)cos(x)-sin(3x)sin(x))
cos(2*2x) / (cos(3x)cos(x) - sin(3x)sin(x))
cos(4x)/cos(3x+x)
cos(4x)/cos(4x)
1
1/2[sin(4pi/3)cos(5pi/6)-(cos(4pi/3)sin(5pi/6))-(sin(4pi/3)cos(5pi/6)-sin(5pi/6)cos(4pi/3))]
1/2[sin(4pi/3)cos(5pi/6)-(cos(4pi/3)sin(5pi/6))-(sin(4pi/3)cos(5pi/6)-sin(5pi/6)cos(4pi/3))]
1/2[sin((4pi/3)+(5pi/6))-sin((4pi/3)-(5pi/6))
cos(4pi/3)sin(5pi/6)
-1/2*1/2=/1/4
12 = 5sin(1/2(x+π))+16
Given the equation above, solve for two possible solutions (between 0 and 2π).
-5 and 8
Basketball player that scored the last points at a basketball game at the old Watertown high school gym?
Who is Mia Caterino?
Prove tan(x) = sin(2x) / (1+cos(2x))
Start with left side:
tan(x)
tan(2x/2)
root((1-cos(2x))/(1+cos(2x))
(root((1-cos(2x))/(1+cos(2x))) * (root(1+cos(2x)) / (root(1+ cos(2x))
(root((1-cos(2x))*(1+cos(2x)))) / (root((1+cos(2x))*(1+cos(2x))))
(root(1+cos(2x)-cos(2x)-cos2(2x))) / (root(1+cos(2x))2)
(root(1-cos2(2x))) / (root(1+cos2x)2)
(root(sin2(2x))) / (root(1+cos(2x)2))
sin(2x) / (1+cos(2x))
(2(1-sin2(19pi/4)-sin2(19pi/4)+cos2(19pi/4)* 1/2sin(37pi/6)cos(37pi/6))-sqrt(1+cos(pi/4)/2)
(2(1-sin2(19pi/4)-sin2(19pi/4)+cos2(19pi/4)* 1/2sin(37pi/6)cos(37pi/6))-sqrt(1+cos(pi/4)/2)
2cos2(3pi/4)-1 * 1/2sin(37pi/6)cos(37pi/6))-sqrt(1+cos(pi/4)/2)
cos(2(3pi/4))-1 * 1/2sin(37pi/6)cos(37pi/6))-sqrt(1+cos(pi/4)/2)
cos(3pi/2)-1 * 1/2sin(37pi/6)cos(37pi/6))-sqrt(1+cos(pi/4)/2)
0-1 * 1/(sin(2(pi/6))-sqrt(1+cos(pi/4)/2)
(-1/sqrt(3)/2)-sin((pi/4)/2)
-2/sqrt3-1
(-2sqrt3/3)-1
Using Diagram 4 on the paper provided, solve for x.
68.22
1st class Mr. Spillane taught at Watertown High School
What is geometry?