Various Trig. Equations?

Question

Find the exact value:
1. cos (-pi/120
2. cos^2+sin^2
3. cos (-pi/12)

Verify:
1. cos{(pi/2)-x}=sinx

Solve in the interval {0, 2pi)
1. cos-2sinx cosx= 0
2. 2sin^2 (x/3)-1= 0

Answer 1

1) cos(-pi/120)
=0.999657324975557

2) cos^2+sin^2
=1

3) cos (-pi/12)
=0.965925826289068
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1) cos (pi/2 - x) = sin x
True.
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1) (cos x) - 2*(sin x)*(cos x) = 0
x = 3pi/2, pi/2, pi/6, 5pi/6

2) (sin^2 x) * (2x/3) - 1 = 0
x = 1.216pi, 0.686pi, 0.480pi (3d.p.)

Answer 2

1. you mean cos(-pi/12)?? the you need to use a half-angle formula with A=pi/6. ALSO cos (-pi/12)=cos (pi/12)

2. you need an arguement for the trig fcn in here... ALWAYS =1.

3. SEE #1.

Verify: use sum/difference formula:
cos(pi/2-x) = cos (pi/2)cosx + sin (pi/2) sin x
exact values .. = (0) cos x + (1) sin x = sin x

Solve:
1. Factor: cosx(1-2sinx)=0. Now two equations: cos x = 0 and 1-2 sin x =0 by zero property rule. ANSWERS: x=pi/2, 3pi/2, pi/6, and 5 pi/6

2. add one to both sides, then divide by two to get:
sin^2 (x/3) = 1/2.
Square root each side:
sin (x/3) = plus or minus 1/sqrt(2)
inverse trig yields
(x/3) = pi/4, 3pi/4, 5pi/4, 7pi/4
multiply by three (keep in domain): 3pi/4 ONLY

Answer 3

3. The sine wave is "based" at (800 + 2 hundred)/2 = 500 with amplitude (800 - 2 hundred)/2 = 3 hundred. So F(t) = 500 + 3 hundred sin(a t + b) the place constants a and b are chosen so as that a million/2 the era is 5.a million - 2.9 = 2.2 years, and we can start up a era at (5.a million + 2.9) / 2 = 4.0. So the kind of foxes F(t) = 500 + 3 hundred sin( 2 pi (t - 4.0) / 4.4 ) for t in years 4. a-d Plug interior the specified values for t interior the end results of section 3.