9 - 1 Solving Right Triangles
 
 
 
Samples of solving right triangles
1)  For the right triangle ABC shown, find the side lengths to three significant digits and angles to tenths place.
 
To find / B, subtract / A from 90.  B = 90 - 25 = 65
/ B = 65o
To find side b, use / B.  We can use:
tan / B = opp/adj = b/40
tan 65 = b/40
40 tan 65 = b
85.8 = b
To find side c, use / A.  We can use:
sin / A = opp/hyp = 40/c
sin 25 = 40/c
c sin 25 = 40
c = 40/sin 25
c = 94.6

2)  Find the missing parts of the triangle shown below:
 
To find / A we can use tan
tan / A = opp/adj
tan / A = 50/30
/ A = tan-1(50/30)
/ A = 59.0o
To find / B, subtract / A from 90
/ B = 90 - 59 = 31o
/ B = 31o
To find c, use the pythagorean theorem.
                                                            ________
c = \/ 302 + 502
c = 58.3

 
3)  From a point 100 meters from the base of a tower, the angle of elevation to its top is 42o.  Find its height to three significant digits.
 
 
To find h, use tangent
tan 38 = opp/adj
tan 38 = h/100
100 tan 38 = h
78.1 = h
The tower is 78.1 meters high.

 
4)  From the top of a lighthouse 54 meters above the sea, the angle of depression of a buoy is 17o.  Find the horizontal distance from the buoy.  Round to three significant digits.
 
To find x, use cotangent.
cot 17 = adj/opp
cot 17 = x/54
54 cot 17 = x
177 = x
177 meters to the buoy.
 
We could also have used the tangent by finding the other acute angle.
90 - 17 = 73
tan 73 = x/54
54 tan 73 = x
177 = x
Your choice!!

 
5)  From the top of a building that overlooks a lake, Jack watches a boat sailing directly toward the building.  He notes that the angle of depression now to the boat is 40o and later it changes to 27o.  His room is 120 feet above the lake.  How far did the boat travel during that period?
 
We need to calculate two distances and then subtract these two.  Both distances we calculate will give us the distance from the boat to the shore at different times.  So, by subtracting we will get the distance the boat traveled.
Calculating the distance to the shore at the first observation of 27o:
Use tangent:
tan 63 = opp/adj
tan 63 = x/120
120 tan 63 = x
235.5 = x
Now calculate the distance at 40o:
Use tangent again
tan 50 = opp/adj
tan 50 = y/120
120 tan 50 = y
143.0 = y
Now subtract y from x and we will have the distance the boat traveled.
235.5 - 143.0 = 92.5
The boat traveled 92.5 feet.

 That should give you some examples of the trig of the right triangle.  Next section deals with calculating the area of a triangle without knowing the height.