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Law of Cosines Worked Examples cheat sheet - grade 10-12

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Math Grade 10-12

Law of Cosines Worked Examples Cheat Sheet

A printable reference covering the Law of Cosines, missing sides, missing angles, triangle setup, and law selection for grades 10-12.

TopicsTrigonometryGeometryAlgebraWord ProblemsMathematicsStudy Guide
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The Law of Cosines is used to solve triangles when the Pythagorean Theorem is not enough. This cheat sheet focuses on worked-example setup for finding a missing side or a missing angle. It helps students identify which sides and angles belong together before substituting values.

It is especially useful for non-right triangles in geometry, trigonometry, and applied measurement problems.

The main formula is c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, where CC is the angle opposite side cc. To find a side, substitute two sides and the included angle, then take the square root. To find an angle, rearrange the formula to cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}, then use inverse cosine.

Choosing between the Law of Cosines and Law of Sines depends on whether the triangle information is SASSAS, SSSSSS, ASAASA, AASAAS, or SSASSA.

Key Facts

  • The Law of Cosines is c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, where side cc is opposite angle CC.
  • To find a missing side with SASSAS, use c=a2+b2−2abcos⁡Cc = \sqrt{a^2 + b^2 - 2ab\cos C}.
  • To find a missing angle with SSSSSS, use cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab} and then C=cos⁡−1(a2+b2−c22ab)C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right).
  • The included angle is the angle between the two known sides, such as angle CC between sides aa and bb.
  • If C=90∘C = 90^\circ, then cos⁡90∘=0\cos 90^\circ = 0, so the Law of Cosines becomes c2=a2+b2c^2 = a^2 + b^2.
  • Use the Law of Cosines first for SASSAS or SSSSSS triangle information.
  • After finding one angle in an SSSSSS triangle, the Law of Sines can often find another angle using sin⁡Aa=sin⁡Bb\frac{\sin A}{a} = \frac{\sin B}{b}.
  • Always check that the largest angle is opposite the longest side and the smallest angle is opposite the shortest side.

Vocabulary

Law of Cosines
A triangle formula, c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, that relates three sides and one included angle.
Included angle
The included angle is the angle formed between two known sides, such as CC between sides aa and bb.
Opposite side
An opposite side is the side across from a given angle, so side cc is opposite angle CC.
SAS
SASSAS means two sides and the included angle are known, which is the standard setup for finding a missing side.
SSS
SSSSSS means all three side lengths are known, which is the standard setup for finding a missing angle.
Inverse cosine
Inverse cosine, written cos⁡−1(x)\cos^{-1}(x), gives the angle whose cosine is xx.

Common Mistakes to Avoid

  • Using the wrong opposite pair is incorrect because cc must be opposite CC in c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C.
  • Forgetting the square root when finding a side is incorrect because the formula gives c2c^2, so the final side length is c=c2c = \sqrt{c^2}.
  • Using the Law of Sines for SASSAS first is usually wrong because SASSAS does not give an opposite side-angle pair.
  • Entering the calculator in radians when the problem uses degrees gives the wrong value because cos⁡60∘\cos 60^\circ is not the same input as cos⁡60\cos 60 radians.
  • Dropping the negative sign in −2abcos⁡C-2ab\cos C changes the formula and can make the computed side much too large or too small.

Practice Questions

  1. 1 In triangle ABCABC, a=7a = 7, b=10b = 10, and C=48∘C = 48^\circ. Find cc using c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C.
  2. 2 In triangle ABCABC, a=8a = 8, b=11b = 11, and c=13c = 13. Find angle CC using C=cos⁡−1(a2+b2−c22ab)C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right).
  3. 3 A triangle has sides 99, 1212, and 1515. Use the Law of Cosines to decide whether the angle opposite side 1515 is acute, right, or obtuse.
  4. 4 Explain why the Law of Cosines is the better first choice than the Law of Sines when you are given two sides and the included angle.

Understanding Law of Cosines Worked Examples

The cosine part of the calculation describes the shape of the triangle, not just its measurements. Imagine two known sides starting from the same corner. When the angle between them is narrow, their far ends stay relatively close together.

When that angle opens wider, the far ends separate. Cosine measures this effect through the direction of one side compared with the other. For an acute angle, the correction reduces the result.

For an obtuse angle, cosine is negative, so the correction increases the result. This gives a quick prediction before any calculator work. The side opposite an obtuse angle should be especially long.

Careful labeling prevents many errors. Use capital letters for corners and the matching lowercase letter for the side directly across from each corner. The labels do not depend on whether a side is drawn at the bottom or on the left of a diagram.

Redraw a crowded triangle if needed. Mark every given length and angle before choosing a method. In a side and angle setup, verify that the known angle touches the two known sides.

A common mistake is using an angle opposite one of those sides instead. That changes the triangle relationship completely, even when the arithmetic is perfect.

Calculator settings matter most when finding an angle. In most school geometry problems, the calculator must be in degree mode. A calculator in radian mode can produce a number that looks reasonable but represents a different unit.

Keep several decimal places during the calculation, then round only at the end. The value entered into inverse cosine must be from negative one to positive one. A value just outside that range can signal early rounding or an incorrect setup.

Before solving a triangle from three sides, check the triangle inequality. Each pair of side lengths must add to more than the remaining side. Otherwise, no triangle exists.

These ideas appear whenever an unknown diagonal or distance crosses a non-right angle. A surveyor can use two measured paths and the angle between them to find the distance between landmarks. A builder can check a sloping brace in a frame.

In physics, two forces acting from the same point can form a triangle, and the angle between their directions affects the combined force. In each setting, units must match before calculation. A strong final check compares size and position.

The longest side must face the widest angle. If a result violates that pattern, return to the diagram, the labels, and the calculator mode before trusting the answer.