4.8. Applications of Basic Triangle Trigonometry http://www.ck12.org
Concept Problem Revisited
Sometimes when using the Law of Sines you can get answers that do not match the Law of Cosines. Both answers
can be correct computationally, but the Law of Sines may involve interpretation when the triangle is obtuse. The
Law of Cosines does not require this interpretation step.
First, use Law of Cosines to find^6 B:
122 = 32 + 142 − 2 · 3 · 14 ·cosB(^6) B=cos−^1
( 122 − 32 − 142
− 2 · 3 · 14
)
≈ 43. 43 ...◦
Then, use Law of Sines to find^6 C. Use the unrounded value forBeven though a rounded value is shown.
sin 43. 43 ◦
12 =sinC
14
14 sin 43. 43 ◦
12 =sinC(^6) C=sin−^1
(14 sin 43. 43 ◦
12
)
≈ 53. 3 ◦
Use the Law of Cosines to double check^6 C.
142 = 32 + 122 − 2 · 3 · 12 ·cosC
C=cos−^1( 142 − 32 − 122
− 2 · 3 · 12
)
≈ 126. 7 ◦
Notice that the last two answers do not match, but they are supplementary. This is because this triangle is obtuse and
the sin−^1
(o p p
hy p)
function is restricted to only producing acute angles.Vocabulary
Angle of elevationis the angle at which you view an object above the horizon.
Angle of depressionis the angle at which you view an object below the horizon. This can be thought of negative
angles of elevation.
Bearingis how direction is measured at sea. North is 0◦, East is 90◦, South is 180◦and West is 270◦.