Obj. 43 Laws of Sines and Cosines

The student is able to (I can):

• Use the Law of Sine and the Law of Cosine to solve problemsproblems

Beyond Right Angle Trigonometry

• When we first started talking about trigonometry, we started off with right triangles and defined the trig functions as ratios between the various sides.

• It turns out we can extend these ratios to all triangles, such that if any three of the six side and angle measures such that if any three of the six side and angle measures of a triangle are known (including at least one side), then the other three can be found.

Law of Sines In any triangle ABC, with sides a, b, and c,

(i.e. the lengths of the sides in a triangle

= =a b c

sinA sinB sinCCB

A

c b

a

(i.e. the lengths of the sides in a triangle are proportional to the sines of the measures of the angles opposite them).

To solve for an angle, we can also write this as

• The law of sines involves two sides and

= =sinA sinB sinC

a b cCB

A

c b

a

• The law of sines involves two sides and the two angles opposite those sides.

• When using the law of sines, a good strategy is to select an equation so that the unknown variable is in the numerator and all other variables are known.

Example Find side u if B = 32.0°, U = 81.8°, and b = 42.9 cm. G

UB g

u 42.9 cm

32.0° 81.8°

B

=

=° °

°=

°

≈

b u

sinB sinU42.9 u

sin32.0 sin81.842.9sin81.8

usin32.0

80.1 cm

Law of Cosines In any triangle ABC, with sides a, b, and c,

c2 = a2 + b2 — 2ab cos C

CB

A

c b

a

• The law of cosines involves three sides and an angle. Set up your equation so that the side opposite the angle is c, and the other two sides are a and b.

Law of Cosines • Example: Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m.

12.9 m a

C

BA15.4 m

42.3°

Example Find side a if A = 42.3°, b = 12.9 m, and c = 15.4 m.

12.9 m a

C

BA 15.4 m

42.3°

( )( )

= + −

= + − °

≈

≈

2 2 2

2 2 2

2

a b c 2bccosA

a 12.9 15.4 2 12.9 15.4 cos42.3

a 109.7

a 10.47 m