Answer:
4.) 263.3 yd
Step-by-step explanation:
Trigonometry Basics
The three basic trig functions are
- [tex]\rm sin(x)=\frac{opposite}{hypotenuse}[/tex]
- [tex]\rm cos(x)=\frac{adjacent}{hypotenuse}[/tex]
- [tex]\rm tan(x)=\frac{adjacent}{opposite}[/tex],
where "opposite", "adjacent", and "hypotenuse" are the side lengths of a right triangle and are determined by the location of the reference angle x.
[tex]\hrulefill[/tex]
Solving the Problem
We're given the triangle's hypotenuse and need to find the length of a side length opposite to the triangle's angle of elevation (bottom left).
This follows the sine function,
[tex]\rm sin(a)=\dfrac{x}{580}[/tex]
but don't know the elevation's measure.
Identifying the Missing Angle
We're given the angle adjacent to the angle of depression (top right) and a horizontal line that looks parallel to the horizontal leg of the right triangle.
If we consider the hypotenuse as the transversal, the angle of elevation and the adjacent angle are alternate interior angles, thus making the missing angle have a measure of 27 degrees.
**Alternate Interior Angles: angles diagonally located on opposite sides of the transversal always have the same angle measure.**
Putting it All Together
So,
[tex]\rm sin(27^\circ)=\dfrac{x}{580}[/tex]
[tex]\rm 580sin(27^\circ)=\boxed{x=263.3}[/tex] .
