Need help with trig graph questions

• Nov 5th 2009, 09:32 AM
BG5965
Need help with trig graph questions
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

• Nov 5th 2009, 11:43 AM
Hello BG5965
Quote:

Originally Posted by BG5965
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

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?