6.11—Law of Sines
6.11 Warm Up
1. What is the third angle measure in a triangle with angles
measuring 65° and 43°?
Find each value. Round trigonometric ratios to the nearest
hundredth and angle measures to the nearest degree.
2. sin 73°
3. cos 18°
4. tan 82°
5. sin-1 (0.34)
6. cos-1 (0.63)
7. tan-1 (2.75)
8.
27
y
6.11—Law of Sines
Objective: Use the Law of Sines to solve triangles.
Examples
1. Use your calculator to find each trigonometric ratio.
Round to the nearest hundredth.
A. tan 103°
B. cos 165°
tan 103°  –4.33
cos 165°  –0.97
C. sin 93°
sin 93°  1.00
You can use the Law of Sines to solve a triangle if you are given
• two angle measures and any side length (ASA or AAS) or
• two side lengths and a non-included angle measure (SSA).
Examples
Find the measure. Round lengths to the nearest
tenth and angle measures to the nearest degree.
2. Find FG.
Law of Sines
Substitute the given values.
FG sin 39° = 40 sin 32°
Cross Products Property
Divide both sides by sin 39.
3.
Find mQ.
Law of Sines
Substitute the given values.
Multiply both sides by 6.
Use the inverse sine function to find
mQ.
4. Find NP.
Law of Sines
Substitute the given values.
NP sin 39° = 22 sin 88°
Cross Products Property
Divide both sides by sin 39°.
5. Find mL.
Law of Sines
Substitute the given values.
10 sin L = 6 sin 125°
Cross Products Property
Use the inverse sine
function to find mL.
6. Find mX.
Law of Sines
Substitute the given values.
7.6 sin X = 4.3 sin 50°
Cross Products Property
Use the inverse sine function
to find mX.
7. Find AC.
mA + mB + mC = 180°
mA + 67° + 44° = 180°
mA = 69°
∆ Sum Thm.
Substitution
Simplify.
Law of Sines
Substitute the given values.
AC sin 69° = 18 sin 67°
Cross Products Property
Divide both sides by sin 69°.
When using the Law of Sines with SSA (ASS), sometimes there are TWO possible answers !
If we know angle A, and sides a and b, sometimes we can swing side a to left or right and
come up with two possible results (a small triangle and a much wider triangle). Both
answers are right!
Just think "could I swing that side the other way to also make a correct answer?"
8. Find mR.
Draw the other possible triangle.
sin 𝑅
sin 𝑄
=
𝑟
𝑞
When angle Q is an acute angle, what
is it about PR that makes two triangles
possible?
sin 𝑅
sin 39
=
41
28
sin 39
sin 𝑅 =
∙ 41
28
sin 𝑅 = 0.9215
The calculator won’t tell you the other angle.
All you need to do is subtract 67.1° from 180°.
𝑅 = sin−1 0.9215
180° − 67.1° = 32.9°
𝑅 = 67.1°
𝑅 = 32.9°