The Law of Sines and the Law of Cosines help you solve triangles that are not right triangles. They connect side lengths and angle measures in any triangle, so they are useful when the Pythagorean theorem or basic sine, cosine, and tangent are not enough. These laws are important in geometry, surveying, navigation, engineering, and map making.
They give you a way to find missing parts of a triangle from limited information.
Understanding Law of Sines & Law of Cosines
The most important habit is matching each side to the angle directly across from it. Start by drawing a clear sketch, even if the picture in a problem looks neat already. Label the vertices with capital letters and place the matching lowercase side across from each angle.
This prevents a common error where a student pairs a side with an angle beside it. A triangle may be turned or flipped without changing these opposite pairs. The labels, not the position on the page, determine which measurements belong together.
Choose a method from the information you have, rather than from the shape of the sketch. When two angles are known, the third angle can be found by subtracting their total from one hundred eighty degrees. This often creates a known angle and its opposite known side, which gives a useful starting pair for the sine relationship.
When two sides meet at a known angle, use the cosine relationship first to find the side across from that angle. When all three sides are known, cosine can find an angle. After finding one missing part, the other law may become the easier tool for completing the triangle.
The cosine relationship extends an idea from right triangles. In a right triangle, the angle between two sides is ninety degrees. The cosine of ninety degrees is zero, so the extra adjustment disappears and the familiar square relationship remains.
For an acute angle, the adjustment subtracts some amount, making the opposite side shorter than it would be in a right triangle with the same two nearby sides. For an obtuse angle, cosine is negative, so the subtraction effectively adds length. This matches the picture because opening the angle wider moves the far endpoints farther apart.
The sine relationship needs extra care in the SSA case, where two sides and an angle not between them are given. The measurements can describe no triangle, one triangle, or two different triangles. This happens because an angle can sometimes be acute or obtuse while having the same sine value.
A calculator may return the acute angle first. Check whether its supplement, found by subtracting the angle from one hundred eighty degrees, could fit with the given angle and still leave a positive third angle. Surveyors and navigators face similar checks when they calculate distances from measured directions.
In class, keep units consistent, round only near the end, and test the final values. Side lengths should match angle sizes, with the largest side opposite the largest angle.
Key Facts
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: c^2 = a^2 + b^2 - 2ab cos C
- Use the Law of Sines when you know ASA, AAS, or sometimes SSA information.
- Use the Law of Cosines when you know SAS or SSS information.
- The angles of any triangle add to 180 degrees: A + B + C = 180 degrees
- In standard triangle notation, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
Vocabulary
- Law of Sines
- A formula that relates each side of a triangle to the sine of its opposite angle.
- Law of Cosines
- A formula that relates the sides of a triangle to the cosine of one included angle.
- Opposite Side
- The side across from a given angle in a triangle.
- Included Angle
- The angle formed between two known sides of a triangle.
- Oblique Triangle
- A triangle that does not have a 90 degree angle.
Common Mistakes to Avoid
- Matching an angle with the wrong side. In the Law of Sines, each angle must be paired with the side directly across from it.
- Using the Law of Sines for SAS information. SAS usually requires the Law of Cosines because the known angle is between two known sides.
- Forgetting to use the 180 degree angle sum. After finding one or two angles, subtract from 180 degrees to find the missing angle.
- Rounding too early in a multi-step problem. Keep extra decimal places until the final answer so the result stays accurate.
Practice Questions
- 1 In triangle ABC, A = 40 degrees, B = 70 degrees, and a = 12 cm. Find angle C, then find side b using the Law of Sines.
- 2 In triangle ABC, a = 8 cm, b = 11 cm, and C = 60 degrees. Find side c using the Law of Cosines.
- 3 A triangle has two known sides and the angle between them. Explain why the Law of Cosines is a better first choice than the Law of Sines.