Need help with trig graph questions

BG5965 · #1 $Old$ November 5th, 2009, 10:32 AM

Hi, there are a few trig questions I'm not sure how to work out:

The equation y = a cos bx can be used to describe some natural cycles, such as tides and the phases of the moon.

1] Determine the constants a and b and write the equation representing the scenario.

a) A 12 hour gap between high tides of 2.3m and low tides of -2.3m
b) Same as a), except tides vary from 0.7m to 5.3m.

2] State the greatest possible value of 2sin(x - 30 deg) and the smallest possible value of x that will get this maximum value.

3] State the period, amplitude and range of this equation: y = -2sin(x - 45 deg) + 2

Thanks in advance for help, and please explain your working. Thanks!

Grandad · #2 $Old$ November 5th, 2009, 12:43 PM

Hello BG5965

Quote:

Originally Posted by BG5965 $View Post$

Hi, there are a few trig questions I'm not sure how to work out:

The equation y = a cos bx can be used to describe some natural cycles, such as tides and the phases of the moon.

1] Determine the constants a and b and write the equation representing the scenario.

a) A 12 hour gap between high tides of 2.3m and low tides of -2.3m
b) Same as a), except tides vary from 0.7m to 5.3m.

2] State the greatest possible value of 2sin(x - 30 deg) and the smallest possible value of x that will get this maximum value.

3] State the period, amplitude and range of this equation: y = -2sin(x - 45 deg) + 2

Thanks in advance for help, and please explain your working. Thanks!

What you need to know about the graph of the equation $y = a\cos bx$ :

Amplitude: $y$ varies between $+a$ and $-a$ . The value of $a$ is called the amplitude of the graph.

Period: the distance between corresponding points on consecutive cycles - for instance two consecutive peak values - is called the period. If $x$ is measured in degrees, the period is $\frac{360}{b}$ .

Initial value. When $x = 0, y = a$ . So the graph starts at its maximum value.

1 a) The 12-hour gap between consecutive high tides gives the value of the period. So using the formula for the period above:

$\frac{360}{b}=12$

$\Rightarrow b = ...$ ?

The maximum and minimum values are $+2.3$ and $-2.3$ , so the value of $a$ is ... ?

b) Here the values vary between $0.7$ and $5.3$ , a difference of $4.6$ . You'll notice that these values are exactly $3$ more than the values in (a). So

the amplitude is still $2.3$ , but

you'll need to add a constant value of $3$ to the equation in (a). Answer: $y = a\cos bx + 3$ (with the same values of $a$ and $b$ as before)

2) The sine function again takes values between $+1$ and $-1$ , with its first maximum occurring at $90^o$ . So $2\sin(x-30^o)$ varies between $+2$ and $-2$ , with the first maximum occurring when $x-30 = 90 \Rightarrow x = ...$ ?

3) Use the same ideas again to get the answers here.

Can you fill in the gaps?

Grandad