How to Use the Law of Cosines to Find Angle Measure?
TL;DR: Here is when the Law of Cosines saves you: you know the lengths of all three sides of a triangle, and you want to find an angle. Plug into c-squared equals a-squared plus b-squared minus 2ab cosine C, rearrange so cosine C is alone, and one inverse-cosine button later you've got the angle. It is the Pythagorean Theorem's smarter cousin. It works on any triangle, not just right ones.
Key takeaways:
- Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C), where C is the angle opposite side c.
- Rearranged for angle: cos(C) = (a^2 + b^2 - c^2) / (2ab).
- Use when you know SSS (all three sides) or SAS (two sides and included angle).
- C is the angle opposite the side you want to solve for (c).
- Always take inverse cosine (cos^-1) at the final step.
If we know the sizes of the three sides of the triangle, we can use the law of cosines to find the size of each angle of the triangle. In this guide, you will learn more about the law of cosines.
Use the Law of Cosines to Find Angle Measure: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Use the Law of Cosines to Find Angle Measure: pop-up practice
Step-by-step guide to using the law of cosines to find angle measure
The law of cosine says that the square of each side of a triangle is equal to the difference between the sum of squares of the other two sides and twice the product of the other sides and the cosine angle included between them.
Let \(a, b,\) and \(c\) be the lengths of the three sides of a triangle and \(A, B,\) and \(C\) be the three angles of the triangle. Then, the law of cosines states that:
- \(\color{blue}{a^2=b^2+c^2-2bc.\:cos\:A}\)
- \(\color{blue}{b^2=\:c^2\:+\:a^2\:-\:2ca·\:cosB}\)
- \(\color{blue}{c^2=\:a^2+\:b^2-\:2ab·\:cosC}\)
If we know the sizes of the three sides of the triangle, we can use the law of cosines to find the size of each angle of the triangle. These formulas can be used to find the cosine of any angle of \(∆\: ABC\):
- \(\color{blue}{cos\:A=\frac{\:b^2+c^2-a^2}{2bc}}\)
- \(\color{blue}{cos\:B=\:\frac{a^2+c^2-b^2}{2ac}}\)
- \(\color{blue}{cos\:C=\:\frac{a^2+b^2-c^2}{2ab}}\)
Using the Law of Cosines to Find Angle Measure , Example 1:
In \(ABC\) triangle, \(a=12,\:b=8,\:c=6\). Find the angle \(B\).
Solution:
Write the law of cosines in terms of \(cos B\): \(cos\:B=\:\frac{a^2+c^2-b^2}{2ac}\)
\(cos\:B=\frac{12^2+6^2-8^2}{2\times 12\times 6}\)
\(cos B =0.8\)
\(B= 36.33^{\circ }\)
Exercises for Using the Law of Cosines to Find Angle Measure
- In \(∆\:ABC\), \(a=25,\:b=10,\:c=18\). Find the angle \(A\).
- In \(∆\:ABC\), \(a=9,\:b=8,\:c=5\). Find the angle \(C\).
- \(\color{blue}{123.94^{\circ \:\:}}\)
- \(\color{blue}{33.55^{\circ }}\)
Frequently Asked Questions
How do you use the law of cosines to find an angle?
Rearrange the formula so the cosine stands alone: cos(C) = (a² + b² − c²) / (2ab), where C is the angle opposite side c. Substitute the three side lengths, simplify to a decimal, then press inverse cosine (cos⁻¹). For sides a = 5, b = 7 and c = 9, cos(C) = (25 + 49 − 81)/70 = −0.1, so C ≈ 95.7°. Keep your calculator in degree mode.
How do you find angle B using the law of cosines?
Use the version of the formula built around side b: b² = a² + c² − 2ac·cos(B), which rearranges to cos(B) = (a² + c² − b²) / (2ac). The side being subtracted is always the one opposite the angle you want. For a = 6, b = 8 and c = 10, cos(B) = (36 + 100 − 64)/120 = 72/120 = 0.6, so B = cos⁻¹(0.6) ≈ 53.1°.
How do you find all the angle measures of a triangle using the law of cosines?
Find one angle with the rearranged formula, find a second angle the same way using a different pair of sides, then subtract both from 180°. For sides a = 7, b = 8 and c = 9: cos(A) = (64 + 81 − 49)/144 ≈ 0.667, so A ≈ 48.2°; cos(B) = (49 + 81 − 64)/126 ≈ 0.524, so B ≈ 58.4°; then C ≈ 180° − 48.2° − 58.4° = 73.4°.
Can the law of cosines be used for any triangle?
Yes. The law of cosines works for acute, right and obtuse triangles, as long as you know all three sides (SSS) or two sides and the angle between them (SAS). When the angle is 90°, cos(90°) = 0 and the formula shrinks to c² = a² + b², the Pythagorean Theorem. That is why it is often described as the general form of the Pythagorean Theorem.
What is the difference between the law of cosines and the law of sines?
Both connect sides and angles in any triangle, but they suit different starting information. The law of cosines, c² = a² + b² − 2ab·cos(C), handles SSS and SAS cases. The law of sines, a/sin(A) = b/sin(B) = c/sin(C), handles AAS, ASA and SSA cases. When you are finding angles from three known sides, start with the law of cosines, because it never gives an ambiguous answer.
What is the law of cosines used to find?
The law of cosines finds a missing side when you know two sides and the included angle, and it finds any angle when you know all three sides. Surveyors, pilots and engineers use it to work out distances and directions they cannot measure directly. A handy habit: find the largest angle (opposite the longest side) with the law of cosines first, since it is the only angle that could be obtuse.
Related Lessons You May Like
- Significance of the unit circle
- How to use right-triangle trigonometry
- Ambiguous case (SSA) in trigonometry
- How to find similar figures
- How to use the Pythagorean Theorem
For a workbook on trigonometry, Trigonometry for Beginners (resource currently unavailable) covers the unit circle, identities, and applications. Pre-Calculus for Beginners is the natural next step.
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