Remember that for sine, cosine and the period is 2pi/b which means in this case, 2pi divided by 1/3, so it won't be 2pi/3.
For secant, it's the inverse of cosine so if you were able to find key points on cosine, they would be similar for sine. The period should be the same as for cosine. The shift works the same way too.
For cotangent, the period is pi/b. (Remember b is whatever is multiplied by x.)
For finding the key points, that's not too difficult. Once you get the real period value following the advice from the first comment, you have 2 of the 5 you need.
I always find it easier to write them out of order... just works easier for mental math.
1st point will be zero, 5th will be the end of the period. 3rd will be halfway in between, 2nd will be halfway between 1st and 3rd, and 4th will be halfway between 3rd and 5th.
So, for example, say your period is 3pi..
1st point: zero 5th: 3pi 3rd: halfway between zero and 3pi is 3/2pi 2nd: halfway between zero and 3/2pi is 3/4pi 4th: halfway between 3/2 and 3pi is 5/4pi
It's just as easy when there's a phase shift, too, you will just have something different to start with instead of zero.
Cotangent has a period of pi, so the formula is pi/b, and secant is 2pi/b. Just remember that when you divide by a fraction, it's the same as multiplying by the reciprocal, so 5pi divided by 1/2pi isn't 2.5pi, it's actually 10pi.
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For secant, it's the inverse of cosine so if you were able to find key points on cosine, they would be similar for sine. The period should be the same as for cosine. The shift works the same way too.
For cotangent, the period is pi/b. (Remember b is whatever is multiplied by x.)
Good luck!
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Example. 1/2 divided by 3/4.
1/2 times 4/3 = 1 times 4 over 2 times 3 = 4/6 = 2/3
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I always find it easier to write them out of order... just works easier for mental math.
1st point will be zero, 5th will be the end of the period. 3rd will be halfway in between, 2nd will be halfway between 1st and 3rd, and 4th will be halfway between 3rd and 5th.
So, for example, say your period is 3pi..
1st point: zero
5th: 3pi
3rd: halfway between zero and 3pi is 3/2pi
2nd: halfway between zero and 3/2pi is 3/4pi
4th: halfway between 3/2 and 3pi is 5/4pi
It's just as easy when there's a phase shift, too, you will just have something different to start with instead of zero.
Cotangent has a period of pi, so the formula is pi/b, and secant is 2pi/b. Just remember that when you divide by a fraction, it's the same as multiplying by the reciprocal, so 5pi divided by 1/2pi isn't 2.5pi, it's actually 10pi.
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