PLEASE HELP ME If 0 < z ≤ 90 and sin(9z − 1) = cos(6z + 1), what is the value of z? z = 3 z = 4 z = 5 z = 6
1 answer:
Answer:
z = 6
Step-by-step explanation:
We know that ...
sin(x) = cos(90 -x)
Substituting (9z-1) for x, this is ...
sin(9z -1) = cos(90 -(9z -1))
But we also are given ...
sin(9z -1) = cos(6z +1)
Equating the arguments of the cosine function, we have ...
90 -(9z -1) = 6z +1
90 = 15z . . . . . . . . . add (9z-1) to both sides
6 = z . . . . . . . . . . . . divide by 15
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<em>Comment on the graph</em>
The attached graph shows 5 solutions in the domain of interest. These come from the fact that the relation we used is actually ...
sin(x) = cos(90 +360k -x) . . . . . for any integer k
Then the above equation becomes ...
90 +360k = 15z
6 +24k = z . . . . . . . . . for any integer k
The sine and cosine functions also enjoy the relation ...
sin(x) = cos(x -90)
sin(9z -1) = cos(9z -1 -90) = cos(6z +1)
3z = 92 . . . . . equating arguments of cos( ) and adding 91-6z
z = 30 2/3
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Answer:
When we have a given function, y = f(x), the general way of graphing the function is finding a lot of coordinate pairs (x, f(x)) and graphing them in a coordinate axis, and then connect them.
So here we have:
p(x) = 2x^(3)+7x^(2)-3x-18
Just let's find the value of p(x) for different values of x, and let's do it.
p(0) = 2*0^(3)+7*0^(2)-3*0-18 = -18
Then the point (0, -18) belongs to the graph
p(1) = 2*1^(3)+7*1^(2)-3*1-18 = 2 + 7 - 3 - 18 = -12
then the point (1, - 12) belongs to the graph.
p(-1) = 2*(-1)^(3)+7*(-1)^(2)-3*(-1)-18 = -2 + 7 + 3 - 18 = -10
then the point (-1, -10) belongs to the graph:
p(2) = 2*2^(3)+7*2^(2)-3*2-18 = 16 + 28 - 6 - 18 = 20
Then the point (2, 20) belongs to the graph
p(-2) = 2*(-2)^(3)+7*(-2)^(2)-3*(-2)-18 = -16 + 28 + 6 - 18 = 0
Then the point (-2, 0) belongs to the graph.
p(-3) = 2*(-3)^(3)+7*(-3)^(2)-3*(-3)-18 = -54 + 63 + 9 - 18 = 0
Then the point (-3, 0) belongs to the graph.
Now that we have 6 points we can graph them and try to connect them.
We also know that this is a cubic polynomial, so we can expect that it changes its direction two times (no more than that) then knowing that we can estimate a shape for the graph, and you will obtain something like the graph of the image below, where I just connected the known points.
