Law of Sines Basic
The Law of Sines compares these elements of a triangle
What are angles and opposite sides?
Write the Law of Cosines formula.
What is c2=a2+b2−2abcosCc^2 = a^2 + b^2 - 2ab\cos Cc2=a2+b2−2abcosC?
Given a=6a = 6a=6, A=45∘A = 45^\circA=45∘, and B=60∘B = 60^\circB=60∘, find angle C.
What is C=75∘C = 75^\circC=75∘?
You are tracking two paths that meet at a point forming a 60° angle. One is 100 m long, the other 150 m. What is the distance between the two ends?
Use Law of Cosines: c=1002+1502−2(100)(150)cos(60∘)=10000+22500−15000=17500c = \sqrt{100^2 + 150^2 - 2(100)(150)\cos(60^\circ)} = \sqrt{10000 + 22500 - 15000} = \sqrt{17500}c=1002+1502−2(100)(150)cos(60∘)=10000+22500−15000=17500
Forgetting to check this can make the ambiguous case give a wrong answer.
What is the number of possible triangles?
Write the Law of Sines formula
What is asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}sinAa=sinBb=sinCc?
This law is especially useful when you know two sides and this angle.
What is the included angle?
Given a=5a = 5a=5, b=7b = 7b=7, and C=90∘C = 90^\circC=90∘, find side ccc.
What is c=52+72=74c = \sqrt{5^2 + 7^2} = \sqrt{74}c=52+72=74?
A surveyor measures two sides of a plot and the angle between them. What law should they use to find the third side?
What is the Law of Cosines?
Using the Law of Cosines when the angle is not included.
What is misapplying the Law of Cosines?
True or False: The Law of Sines only works on right triangles.
What is false?
Use the Law of Cosines to find side ccc if a=5a = 5a=5, b=7b = 7b=7, and C=60∘C = 60^\circC=60∘.
What is c=52+72−2(5)(7)cos(60∘)=25+49−70(0.5)=39c = \sqrt{5^2 + 7^2 - 2(5)(7)\cos(60^\circ)} = \sqrt{25 + 49 - 70(0.5)} = \sqrt{39}c=52+72−2(5)(7)cos(60∘)=25+49−70(0.5)=39?
Given A=30∘A = 30^\circA=30∘, B=60∘B = 60^\circB=60∘, and c=12c = 12c=12, find side aaa.
Use Law of Sines: a=12⋅sin(30∘)sin(90∘)=6a = \frac{12 \cdot \sin(30^\circ)}{\sin(90^\circ)} = 6a=sin(90∘)12⋅sin(30∘)=6
A lighthouse is visible from two ships. The angle between the ships from the lighthouse is 45°, and the ships are 6 and 8 km away. How far apart are the ships?
Use Law of Cosines: c=62+82−2(6)(8)cos(45∘)c = \sqrt{6^2 + 8^2 - 2(6)(8)\cos(45^\circ)}c=62+82−2(6)(8)cos(45∘)
Assuming triangle rules apply the same for obtuse angles without adjusting this function.
What is cosine?
In triangle ABC, if a=10a = 10a=10, A=30∘A = 30^\circA=30∘, and B=60∘B = 60^\circB=60∘, use the Law of Sines to find bbb.
What is b=10⋅sin(60∘)sin(30∘)=10⋅3/21/2=103b = 10 \cdot \frac{\sin(60^\circ)}{\sin(30^\circ)} = 10 \cdot \frac{\sqrt{3}/2}{1/2} = 10\sqrt{3}b=10⋅sin(30∘)sin(60∘)=10⋅1/23/2=103?
Rearranged, this version of the Law of Cosines allows you to solve for an angle.
What is cosC=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}cosC=2aba2+b2−c2?
In triangle ABC, a=8a = 8a=8, b=9b = 9b=9, and c=10c = 10c=10. Find angle C.
Use Law of Cosines: cosC=82+92−1022(8)(9)=64+81−100144=45144\cos C = \frac{8^2 + 9^2 - 10^2}{2(8)(9)} = \frac{64 + 81 - 100}{144} = \frac{45}{144}cosC=2(8)(9)82+92−102=14464+81−100=14445
A plane travels 300 km, turns 40°, and flies 200 km more. Find the straight-line distance from start to end.
Use Law of Cosines
Incorrect calculator setting for sine/cosine functions.
What is using radians instead of degrees (or vice versa)?
When applying the Law of Sines, this type of triangle can lead to two possible solutions.
What is an SSA triangle (ambiguous case)?
True or False: The Law of Cosines is required when all three sides of a triangle are known.
What is true?
Given triangle ABC with A=40∘A = 40^\circA=40∘, a=10a = 10a=10, and b=12b = 12b=12, determine if it’s ambiguous.
Use Law of Sines to check if more than one triangle fits: Yes, it may be ambiguous (SSA case).
You measure a triangle in a field and know two angles and one side. Which law do you use to find another side?
What is the Law of Sines?
True or False: You can always use Law of Sines when given two sides and an angle.
What is false? (Only safe with ASA or AAS, not SSA)