Question 1:
The sine of the angle is the side length opposite of the angle divided by the hypotenuse.
The side length opposite of 41 degrees is the height of the balloon.
We can multiply the hypotenuse by the sine of 41 degrees to get our answer.
[tex]1053*sin(41)[/tex]
[tex]\approx 986[/tex]
Your answer is the first choice.
Question 2:
In this question, we can imagine an upside down right triangle with an angle of 18 degrees.
The leg opposite of the angle with 18 degrees is the height of the crown.
The length of the other leg is the distance between the base and the ship that we want to find.
The tangent is the opposite leg divided by the adjacent leg. Let's divide 250 by the tangent of 18 degrees to get our answer.
[tex] \dfrac{250}{tan(18)} \approx 769[/tex]
The last choice is your answer.
Question 3:
We are given the lengths of the two legs. Thus, the tangent of the unknown angle is the proportion of those two legs. We are looking for the angle where the opposite side of that angle is the height of the flagpole.
Let x be the unknown angle.
[tex]tan(x)=\dfrac{282}{137}[/tex]
[tex]x=tan^{-1}(\dfrac{282}{137})[/tex]
Use a calculator. (if you don't have a tan^(-1) button, you can use the arctan button)
[tex]x \approx 64[/tex]
Thus, the last choice is our answer. I hope this helps! :)
