http://www.ck12.org Chapter 4. Basic Triangle Trigonometry
Solution:This problem must be done in two parts. First apply the Law of Cosines to determine the length of side
m. This is a SAS situation like Example B. Once you have all three sides you will be in the SSS situation like in
Example A and can apply the Law of Cosines again to find the unknown angleN.
c^2 =a^2 +b^2 − 2 ab·cosC
m^2 = 382 + 402 − 2 · 38 · 40 ·cos 93◦
m^2 ≈ 3203. 1 ...
m≈ 56. 59 ...Now that you have all three sides you can apply the Law of Cosines again to find the unknown angleN. Remember
to match angleNwith the corresponding side length of 38 inches. It is also best to storeminto your calculator and
use the unrounded number in your future calculations.
c^2 =a^2 +b^2 − 2 ab·cosC
382 = 402 +( 56. 59 ...)^2 − 2 · 40 ·( 56. 59 ...)·cosN
382 − 402 −( 56. 59 ...)^2 =− 2 · 40 ·( 56. 59 ...)·cosN
382 − 402 −( 56. 59 ...)^2
− 2 · 40 ·( 56. 59 ...) =cosN
N=cos−^1( 382 − 402 −( 56. 59 ...) 2
− 2 · 40 ·( 56. 59 ...)
)
≈ 42. 1 ◦
Concept Problem Revisited
A triangle that has sides 11, 12 and 13 is not going to be a right triangle. In order to solve for the missing angle you
need to use the Law of Cosines because this is a SSS situation.
c^2 =a^2 +b^2 − 2 ab·cosC
112 = 122 +( 13 )^2 − 2 · 12 · 13 ·cosC
C=cos−^1( 112 − 122 − 132
− 2 · 12 · 13
)
≈ 52. 02 ...◦
Vocabulary
TheLaw of Cosinesis a generalized Pythagorean Theorem that allows you to solve for the missing sides and angles
of a triangle even if it is not a right triangle.
SSSrefers to Side, Side, Side and refers to a property of congruent triangles in geometry. In this case it refers to the
fact that all three sides are known in the problem.
SASrefers to Side, Angle, Side and refers to a property of congruent triangles in geometry. In this case it refers to
the fact that the known quantities of a triangle are two sides and the included angle.
Included angleis the angle between two sides.