Geometry: An Interactive Journey to Mastery

Example 1
$SODQHLVÀ\LQJDWDQDOWLWXGHRIIHHW)URPWKHJURXQG
you spy the plane at an angle of elevation of 57°. What is x, your
KRUL]RQWDOGLVWDQFHWRWKHSODQH"ͼ௘6HHFigure 17.1௘ͽ
Solution
In this problem, we have a right triangle for which we are given information about the side opposite
to the angle of 57°, and we seek information about the side adjacent to it.
This suggests that we look at the ratio oppadj tan 57°.
A calculator gives tan 57° |1.539, so^7000 x |1.539, giving x | 4545 feet.
Example 2
According to Figure 17.2, what are sin xx, cos , and tan x,
each to two decimal places?
Solution
VLQ xx |42.372.1  FRV 72.172.1^22 42.3 | WDQ x ||cossin xx 0.590.81 
Study Tip
x :KHQVROYLQJULJKWWULDQJOHWULJRQRPHWU\SUREOHPVDOZD\VDVNWKHIROORZLQJTXHVWLRQV:KLFKVLGH
OHQJWKVGRZHNQRZ":KLFKVLGHOHQJWKGRZHVHHN"7KHQLWEHFRPHVFOHDUZKLFKRIWKHWKUHHUDWLRV²
VLQHFRVLQHRUWDQJHQW²ZRXOGEHPRVWXVHIXOIRUVROYLQJWKHSUREOHP
Pitfall
x In trigonometry, only angles different from the right angle in the right triangle can be considered.
Although sin 90° , IRUH[DPSOHKDVPHDQLQJLQWKHWKHRU\RIFLUFOHRPHWU\ͼ௘DVWDUZLWKDQ
DQJOHRIHOHYDWLRQRIƒLVGLUHFWO\RYHUKHDGDQGWKHUHIRUHKDVKHLJKW௘ͽLQWKHFRQWH[WRIULJKW
triangles, sin 90° does not make sense: One cannot have a right triangle with a second right angle.
When working with right triangles, one can only focus on the angles with measures strictly between
0° and 90°.

42.3

x 72.1

Figure 17.2

57° x

7000 ft

Figure 17.1