Hello BG5965 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

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

What you need to know about the graph of the equation $\displaystyle y = a\cos bx$:*Amplitude*: $\displaystyle y$ varies between $\displaystyle +a$ and $\displaystyle -a$. The value of $\displaystyle a$ 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 $\displaystyle x$ is measured in degrees, the period is $\displaystyle \frac{360}{b}$.

*Initial value*. When $\displaystyle x = 0, y = a$. 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:$\displaystyle \frac{360}{b}=12$

$\displaystyle \Rightarrow b = ...$ ?

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

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

- the amplitude is still $\displaystyle 2.3$, but

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

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

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

Can you fill in the gaps?

Grandad