**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 = 65^{o}
**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 = 59.0^{o}
**To find **__/__ B,
subtract
__/__ A from 90
__/__ B = 90 - 59 =
31^{o}
__/__ B = 31^{o}
**To find c, use the
pythagorean
theorem.**
**
________**
**c = \/ 30**^{2}
+ 50^{2}
**c = 58.3**

**3) From a point 100
meters
from the base of a tower, the angle of elevation to its top is 42**^{o}.
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 17**^{o}.
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
40**^{o} and later
it changes
to 27^{o}.
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 27**^{o}:
**Use tangent:**
**tan 63 = opp/adj**
**tan 63 = x/120**
**120 tan 63 = x**
**235.5 = x**
**Now calculate the
distance
at 40**^{o}:
**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.**
** **