# Trigonometric Equations - Year 12

Solve the equation -6cos(𝝅t/6)+10 ≥ 12 for t.

For context: The equation models a harbour on a certain day, in which y≥12 represents where it is safe for a ship to enter or leave the harbour. I need to find between which times when the ship can safely enter or leave the harbour, represented by t. (low tide comes at 4am at 4 metres, high tide occurs at 10am at 16 metres).

Therefore, I need four solutions for t.

By simplifying the equation I get cos(𝝅t/6) ≤ -1/3. This is as far as I have gotten, I know I need to take the inverse of the cos and that cos is negative in the 2nd and 3rd quadrants. What comes next?

• Would this be correct? cos(pi t/6) ≤ -1/3 is true when 7/6π ≤ πt/6 ≤ 11/6π or 19/6π ≤ πt/6 ≤ 23/6π So there are two intervals, and solve for t in both intervals so that: 7≤t≤11 and 19≤t≤23 so then the intervals are 7-11am and 7-11pm? I feel like I did this wrong. I made a graph https://imgur.com/a/sJLdBSX could I not just use the x intercepts here.

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• Can you explain more in depth where you got values for t.

• They are approximate, just divide the endpoints of the intervals by pi/6

• Sorry, could you explain what the endpoints of the intervals are? I just can't visualise it at all.

• They are pi +- theta and 3 pi +- theta, as derived above

• I am sorry, I still don't understand. I haven't done this in a while.

• pi is 3.14 and theta is 1.23. Just plug these in and you will get the result

• What should it look like in my calculator?

• I edited the solution and it should be clear now.

• thanks, so I have t but if my graph starts at 4am, as in 4am is 0 on the x axis. Like in the graph https://imgur.com/a/sJLdBSX, how would I change t to get correct times? or would it stay the same?

• https://imgur.com/undefined sorry that graph doesn't show the x axis here is a better one

• If x=0 is 4 am then just add 4 to all the numbers above and it will give you the times, so it will be from 7.65 am to 12.35 pm and from 19.65 pm to 24.35 i.e. 0.35 am

The answer is accepted.
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