# Dumbfounded by this trigonometric equation.

1. Oct 5, 2012

1. The problem statement, all variables and given/known data

A beach ball is riding the the waves near Tofino, BC. The ball goes up and down with the waves according to the formula h= 1.4 sin(∏t/3), where h is the height in meters above sea level, and t is the time in seconds.

A) In the first 10s when is the ball @ sea level.
B) When does the ball reach its greatest height above sea level? Give the first time this occurs and then write an expression for every time the maximum occurs.
C) According to the formula, what is the most the ball goes below sea level.

3. The attempt at a solution

Only way I can imagine solving this would be graphing it with technology, and only for A by setting H to 0.

How can I solve this without technology and how would I find the max and min. for B and C.

2. Oct 5, 2012

### SammyS

Staff Emeritus
What is the period of sin(θ) ?

For what value(s) of θ is sin(θ) = 0 ?

What is the maximum value sin(θ) ever attains. For what value(s) of θ does sin(θ) attain that maximum value?
...

3. Oct 5, 2012

### Mentallic

For A) It's at sea level when h=0, and you want to know the time so you should be solving for t. Can you do that? It's simple trigonometry.
For B) think about the max and min values a sine wave can have, that is, what is the max and min of $y=\sin(x)$, so therefore, what is the max and min of $y=A\sin(x)$ ?
C) is similar to B).

4. Oct 5, 2012

A) 0 correct ?
B/C) Iv never seen a Sin wave until I just had google graph it for me. Max sin = 1 and Min. sin = -1. And Sinθ=0 when θ=0,180. However how does this help me solve the overall b/c w.o technology.

5. Oct 5, 2012

With max sin=1 and min = -1 how do I set up the equation ?

6. Oct 5, 2012

Also for A how do I find the other values for which h=0.

7. Oct 5, 2012

### Mentallic

If you've never seen a sine wave before today, what are you doing answering these questions? And maybe the title should have a been a little different, because it suggests you already know trig but you were just put off by a harder problem.

Anyway, yes, the sine wave oscillates between 1 and -1. This means that if we had the graph $y=\sin(x)$ the max would be 1 and the min would be -1. Thus the graph $y=2\sin(x)$ will oscillate between 2 and -2 because whatever the value of sin(x) is, the y value will be doubled.
In general, $y=A\sin(x)$ will oscillate between A and -A.

The period of a sine wave means how far along the x axis (or time in your case) the sine wave takes to finish a complete cycle. If we're at the maximum value of 1, then for $y=\sin(x)$ it'll take another 360o to get back to 1 again. So sin(x)=1 occurs at x = 180o, 540o, 900o, -180o etc. going on forever. The general formula is $x=180^o+n\cdot 360^o$ for any integer n.

You need to be careful though! Just because the period of the sine wave is 360o doesn't mean when you solve $\sin(x)=0$ it'll happen every 360o because if the wave starts at 1 and by the time it gets back to 1 again, it would have crossed the y-axis twice (went down to -1 then back up again). The general formula for $\sin(x)=0$ is $x=n\cdot 180^o$

Finally, similarly to how you multiply the sine wave by a constant and the max/min changes from 1/-1, if you multiply the argument of the sine by a constant, such as $y=\sin(bx)$ then the period of the sine wave will change. If b=2, then the period is halved which means the graph will oscillate twice as fast (it'll only take 180o to get from 1 back to 1 again).