y = 8 + 7 cos 3t + 7 cos 6t #function #trig

Question

The vertical displacement of the end of a robot arm (in cm) at time t (in [16 marks]
seconds) is given by

y = 8 + 7 cos 3t + 7 cos 6t:

(a) Find all times, t > 0, (in exact form i.e. in terms of fractions of ) 14 marks
where the vertical displacement is 8 cm, i.e. y = 8.

(b) What is the rst time, t > 0, that the vertical displacement is 8 cm? 2 marks
Give your answer exactly and to 2 decimal places.

Answer 1

I'm going to assume you're using degrees for this problem, not radians. However, remember that you can convert from degrees to radians by multiplying the degree measure by π/180. You should definitely check to see whether your teacher wants degrees or radians, as that will change your answer.

 

a)

 

Since we're looking for t when y=8, plug y=8 in to the equation and simplify:

 

8 = 8 + 7cos3t + 7cos6t                        SUBTRACT 8 FROM EACH SIDE

0 = 7cos3t + 7cos6t                              FACTOR 7 OUT OF THE RIGHT SIDE

0 = 7(cos3t + cos6t)                             DIVIDE EACH SIDE BY 7

0 = cos3t + cos6t

 

Now it will help to use the Double Angle Formula for cosine: cos2θ = 2cos²θ - 1

 

Here, cosθ = cos3t and cos2θ = cos6t, since 6t is twice 3t, so:

 

cos6t = 2cos²3t - 1

 

Plug that value in for cos6t in the simplified equation:

 

0 = cos3t + 2cos²3t - 1 = 2cos²3t + cos3t - 1

 

This equation is actually a quadratic equation, which you can see by making a substitution. Replace cos3t with x:

 

0 = 2x² + x - 1

 

This quadratic factors to 0 = (2x - 1)(x + 1). Solving for x, you get:

 

2x - 1 = 0

2x = 1

x = 1/2

 

OR

 

x + 1 = 0

x = -1

 

Now, let's undo that substitution from earlier by replacing x with cos3t:

 

cos3t = 1/2 OR -1

 

Let's start by finding t when cos3t = 1/2. Cosine is 1/2 at 60º and at 300º. Since the cosine function is periodic, each of these values will repeat every 360 seconds, so we can write:

 

3t = 60 + 360n AND 300 + 360n, where n is any integer

 

Dividing both sides by 3 gives you t = 20 + 120n and t = 100 + 120n.

 

Now let's find t when cos3t = -1. Cosine is -1 at 180º. Again, this value will repeat every 360 seconds, so we can write:

 

3t = 180 + 360n, where n is any integer

 

Dividing both sides by 3 gives you t = 60 + 120n.

 

Let's look for a moment at all the possible values of t. Notice that the first equation for t tells you the arm will be displaced 8 cm at 20 seconds, 140 seconds, 260 seconds, etc. The second equation for t gives you values of 100 seconds, 220 seconds, 340 seconds, etc. The third equation for t gives you values of 60 seconds, 180 seconds, 300 seconds, etc. Overall, then, the arm will be displaced 8 cm at 20, 60, 100, 140, 180, 220, 260, 300, 340, etc. seconds, or every 40 seconds starting at 20. So we can condense the equations for t down to a single equation:

 

t = 20 + 40n, where n is any integer (DEGREES)

t = π/9 + 2πn/9, where n is any integer (RADIANS)

 

b)

 

Based on the equation for t, the first time (I think that's what you meant to type) that the arm will be displaced 8 cm is at 20 seconds or π/9 seconds (0.35 seconds), depending on whether your problem asks for answers in degrees or radians.

Answer 2

8 = 8+7*cos3t+7*cos6t

 

=>  cos3t+cos6t = 0

 

=» cos3t+2(cos3t)²-1 = 0

 

So

 

cos3t = 1/2 or -1

 

3t = π/3 or π

 

t = π/9 or π/3