A weight is oscillating on the end of a spring. The position of the weight relative to the point of equilibrium is given by y=1/12(cos8t-3sin8t), where y is the displacement (in meters) and t is the time (in seconds). Find the times when the weight is at the point of equilibrium (y=0) for 0<=t<=1.

1/12(cos8t-3sin8t) = 0

cos8t-3sin8t = 0
3sin8t = cos8t
sin8t/cos8t = 1/3
tan 8t = 1/3
8t = .32175 or 8t = π+.32175
t = .0402 or t = .4329

cos 8t = 3 sin 8 t

tan 8t = sin 8t/cos 8t = .333333333333333
8 t = 18.4 degrees
t = 2.3 in degrees
t = 2.3 *pi/180 = .04014 seconds if you do trig in radians
===================
check t = .04014
8 t = .321 radians or 18.4 deg
cos 18.4 = .949
3 sin 18.4 = .947 pretty close

To find the times when the weight is at the point of equilibrium (y=0), we need to solve the equation y=0.

The given equation is:

y = (1/12)(cos(8t) - 3sin(8t))

Setting y equal to 0, we have:

0 = (1/12)(cos(8t) - 3sin(8t))

Multiplying both sides by 12 to eliminate the fraction, we get:

0 = cos(8t) - 3sin(8t)

Next, we can use trigonometric identities to simplify this equation. In particular, we can use the identity cos(A - B) = cos(A)cos(B) + sin(A)sin(B) to rewrite the equation in terms of a single trigonometric function.

Using the identity cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we can rewrite cos(8t - 90°) as:

cos(8t - 90°) = cos(8t)cos(90°) + sin(8t)sin(90°)

Since cos(90°) = 0 and sin(90°) = 1, this becomes:

cos(8t - 90°) = 0cos(8t) + 1sin(8t) = sin(8t)

Now let's rewrite the equation using this substitution:

0 = cos(8t) - 3sin(8t)
0 = cos(8t - 90°) - 3sin(8t)
0 = sin(8t) - 3sin(8t)

Combining like terms, we have:

0 = -2sin(8t)

To find the values of t that satisfy this equation, we can solve for t by taking the inverse sine of both sides:

sin(8t) = 0

8t = arcsin(0)

Since the sine function is 0 at 0°, 180°, 360°, etc., we can set 8t equal to these values and solve for t:

8t = 0°, 180°, 360°, ...

t = 0°/8, 180°/8, 360°/8, ...

Simplifying, we get:

t = 0, 22.5, 45, 67.5, 90, 112.5, 135, 157.5

However, we only want the values of t between 0 and 1. Let's select the valid values from the above list:

t = 0, 0.25, 0.5, 0.75

Therefore, the times when the weight is at the point of equilibrium (y=0) for 0<=t<=1 are t = 0, 0.25, 0.5, and 0.75.

To find the times when the weight is at the point of equilibrium (y = 0) for the given equation y = (1/12)(cos(8t) - 3sin(8t)), we can set the equation equal to zero and solve for t.

Let's set y = 0:
0 = (1/12)(cos(8t) - 3sin(8t))

Now, let's solve for t. Since we have both cos(8t) and sin(8t), we can use a trigonometric identity to simplify the equation.

The trigonometric identity we can use in this case is:

sin^2(x) + cos^2(x) = 1

Rearranging this identity, we get:

sin^2(x) = 1 - cos^2(x)

Substituting this into our equation:

0 = (1/12)(1 - cos^2(8t) - 3sin(8t))

Now, let's simplify the equation further:

0 = (1/12)(1 - cos^2(8t) - 3sin(8t))
0 = (1/12)(1 - cos^2(8t) - 3sin(8t))
0 = (1/12) - (1/12)cos^2(8t) - (1/4)sin(8t)

Now, let's solve for t by setting each term equal to zero:

(1/12) = 0

This term is a constant and is not dependent on t. So, this term does not affect the solution for t.

-(1/12)cos^2(8t) = 0

To solve this term, we can set cos^2(8t) equal to zero:

cos^2(8t) = 0

Taking the square root of both sides:

cos(8t) = 0

Now, we can solve for t by taking the inverse cosine (cos^-1) of both sides:

8t = cos^-1(0)

Since cos(π/2) = 0, we can write the solution as:

8t = π/2

Now, solving for t:

t = π/16

Next, let's solve for t in the remaining term:

-(1/4)sin(8t) = 0

Setting sin(8t) equal to zero:

sin(8t) = 0

This occurs when the angle 8t is equal to zero or a multiple of π:

8t = 0, π, 2π, 3π, ...

Solving for t in each case:

t = 0/8, π/8, 2π/8, 3π/8, ...

Simplifying the fractions:

t = 0, π/8, π/4, 3π/8, ...

Now, let's check the solution interval 0 ≤ t ≤ 1 to see which values fall within this range:

0 ≤ t ≤ 1:

t = 0 (Valid)
t = π/8 (Valid)
t = π/4 (Valid)
t = 3π/8 (Valid)
t = π/2 (Not valid, as it is outside the range)

Therefore, the times when the weight is at the point of equilibrium (y = 0) for 0 ≤ t ≤ 1 are:
t = 0, π/8, π/4, and 3π/8.