Geometry Law of Sines and Law of Cosines Lesson (Free PDF)
This law of sines cosines lesson for Geometry covers law of sines and law of cosines (Common Core G-SRT.D.9, G-SRT.D.10, G-SRT.D.11), with a free PDF to print.
The Law of Sines says a/sin A = b/sin B = c/sin C, and the Law of Cosines says c squared = a squared + b squared – 2ab cos C. Use sines for AAS and ASA, cosines for SAS and SSS. Area equals one half the product of two sides and the sine of the included angle.
Geometry · Common Core State Standards: G-SRT.D.9, G-SRT.D.10, G-SRT.D.11
Key facts in this law of sines cosines lesson
- Area = (1/2) a b sin C.
- Law of Sines: a/sin A = b/sin B = c/sin C.
- Law of Cosines: c squared = a squared + b squared – 2ab cos C.
- AAS or ASA: use sines. SAS or SSS: use cosines.
- When the included angle is 90 degrees the Law of Cosines becomes the Pythagorean theorem.
How do you find the area of a triangle with two sides and the included angle?
Drop an altitude from one vertex. The height is \(b \text{sin}\,C\), so the area is one half base times height:
\[\text{Area} = \frac{1}{2}ab \text{sin}\,C\]
What is the Law of Sines?
In any triangle, each side divided by the sine of its opposite angle gives the same number.
\[\frac{a}{\text{sin}\,A} = \frac{b}{\text{sin}\,B} = \frac{c}{\text{sin}\,C}\]
Use it when you know two angles and a side (AAS or ASA), or two sides and an angle opposite one of them.
What is the Law of Cosines?
The Law of Cosines extends the Pythagorean theorem to any triangle:
\[c^{2} = a^{2} + b^{2} – 2ab \text{cos}\,C\]
Use it when you know two sides and the included angle (SAS) to find the third side, or all three sides (SSS) to find an angle. When \(C = 90^{\circ}\), the cosine term is 0 and you get \(c^{2} = a^{2} + b^{2}\).
| Given | First step |
|---|---|
| AAS or ASA | Law of Sines (find the third angle first) |
| SAS | Law of Cosines for the third side |
| SSS | Law of Cosines for an angle |
| SSA | Law of Sines (watch for two triangles) |
How do you solve a non-right triangle?
- Sketch the triangle and label the known parts.
- Match the given pattern (AAS, SAS, SSS) to a law.
- Substitute, solve, and round at the end.
- Check that the angles add to 180° and the longest side faces the largest angle.
Example: Sides 7 and 9 have an included angle of \(60^{\circ}\). Then \(c^{2} = 49 + 81 – 2 \cdot 7 \cdot 9 \cdot 0.5 = 67\), so \(c = 8.2\).
Law of sines cosines lesson: questions and answers
When do you use the Law of Sines?
Use it when you know two angles and a side, or two sides and an angle opposite one of them. You need a matching side and opposite angle pair.
When do you use the Law of Cosines?
Use it when you know two sides and the included angle (SAS) or all three sides (SSS). It also works for right triangles.
How do you find the area of a triangle without the height?
Use area = (1/2) a b sin C with two sides and their included angle. Two sides of 8 and 11 with a 30 degree angle give 22 square units.
Why does the Law of Cosines contain a cosine term?
The term corrects the Pythagorean theorem for the angle between the sides. At 90 degrees the cosine is 0 and the term disappears.
How do you check an answer to a triangle problem?
Make sure the angles add to 180 degrees and that the longest side is opposite the largest angle.
How to use this law of sines cosines lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the law of sines cosines lesson as a PDF with the button below and keep it with your notes.
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