Solving right triangles
To solve a triangle is to find all its sides and all its angles. To do it in a right triangle, we have these formulas :
 Example 1:


NOTE #1: To solve a triangle we need 3 data (they mustn’t be 3 angles)
NOTE #2: To find the angle when we have the trigonometric ratio, we use the sin^{1}, cos^{1} and tan^{1} keys in the calculator, pressing SHIFT or INV + sin, cos or tan.
 Example 2:


 Example 3: What is the angle of a 100% slope?

 Example 4: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base, a surveyor measures the angle to the top of the flagpole as 48°. How tall is the flagpole?

Let a denote the height of the flagpole. Then a / 50 = tan 48^{0}, so a = 50 tan 48^{0} ≈ 55.5 ft 
 Example 5: Calculate the height of a tree knowing that, from a point on the ground, the top of the tree can be seen at an angle of 30^{o} and from 10 m closer, the top can be seen at an angle of 60°.
 Example 6: This photo was taken from a point about 500 m horizontally from the Opera House and we observe the waterline below the highest sail as having an angle of depression of 8°. How high above sea level is the highest sail of the Opera House?


This is a simple tan ratio problem.
tan 8° = h/500 so h = 500 tan 8° = 70.27 m.
So the height of the tallest point is around 70 m.
[The actual height is 67.4 m.]
 Example 7: The length of the side of a regular octagon is 12 m. Find the radii of the inscribed and circumscribed circles.


Exercises:
1. The elevation angle of the Sun is 35^{o}. Calculate the length of the shadow of a man of 1.75 m height.
2. Calculate the height of a ladder of 4.5 m that rests against a wall, if the angle between the ladder and the floor is 67^{o}.
Solutions: 1) 2.5 m; 2) 4.14 m
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