# 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