This is a might tricky. First you have to find the altitude. You have to determine if this is a real triangle.
Sin(22) = opposite / hypotenuse. The hypotenuse = 111. The angle is 22
opposite = hypotenuse * sin(22)
opposite = 111 * sin(22)
opposite = 41.58 and this is the altitude.
What have you learned?
Since 42 is larger than 41.58 you have 2 solutions to the triangle. One of the angles is acute, and the other one is obtuse. They are supplementary angles.
Sin(C) / c = Sin(A) / a
Sin(C) = c * Sin(22) / 42
Sin(C) = 111*sin(22)/ 42
Sin(C) = .99003
Sin(C) = 81.9
So that's your first answer. The second answer comes from Finding the supplement to this angle
Let the speed of the wind be , and the speed of the plane in still air be . It takes at least two equations to find the exact solutions to a system of two variables.
Information in this question gives two equations:
It takes the plane three hours to travel from Ottawa to with a tail wind (that is: at a ground speed of .)
It takes the plane four hours to travel from Halifax back to Ottawa while flying into the wind (that is: at a ground speed of .)
Create a two-by-two system out of these two equations:
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There can be many ways to solve this system. The approach below avoids multiplying large numbers as much as possible.
Note that this system is equivalent to
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Either adding or subtracting the two equations will eliminate one of the variables. However, subtracting them gives only on the right-hand side. In comparison, adding them will give , which is much more complex to evaluate. Subtracting the second equation () from the first () will give the equation
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Substitute back into either equation or of the original system. Solve for to obtain .