How to Use the Law of Cosines to Find Angle Measure?

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 \(B\).

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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

1. In \(∆\:ABC\), \(a=25,\:b=10,\:c=18\). Find the angle \(A\).
2. In \(∆\:ABC\), \(a=9,\:b=8,\:c=5\). Find the angle \(C\).

1. \(\color{blue}{123.94^{\circ \:\:}}\)
2. \(\color{blue}{33.55^{\circ }}\)

Frequently Asked Questions

What is the Law of Cosines?

A formula relating the three sides and one angle of any triangle: c^2 = a^2 + b^2 – 2ab cos(C), where C is the angle opposite side c. It generalizes the Pythagorean Theorem (which is the special case C = 90 degrees).

How does it differ from the Pythagorean Theorem?

Pythagorean Theorem works only for right triangles. Law of Cosines works for ANY triangle. When C = 90°, cos(C) = 0 and the formula reduces to c^2 = a^2 + b^2.

How do I rearrange to solve for the angle?

Move the cosine term to one side: cos(C) = (a^2 + b^2 – c^2) / (2ab). Then C = cos^-1((a^2 + b^2 – c^2)/(2ab)).

Walk through an example.

Triangle with sides a=5, b=7, c=9. To find angle C opposite c: cos(C) = (25 + 49 – 81)/(2*5*7) = -7/70 = -0.1. So C = cos^-1(-0.1) ≈ 95.7°.

When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you know SSS (all three sides) or SAS (two sides and the included angle). Use the Law of Sines when you know AAS, ASA, or SSA.

Can the Law of Cosines give an obtuse angle?

Yes — if cos(C) is negative, then C is obtuse (between 90° and 180°). That is one of the law’s strengths: it distinguishes between acute and obtuse triangles cleanly.

What if the side I want is not opposite the angle?

Relabel. The convention is that side c is opposite angle C. You can apply the law in any of three forms: a^2 = b^2 + c^2 – 2bc cos(A), b^2 = a^2 + c^2 – 2ac cos(B), c^2 = a^2 + b^2 – 2ab cos(C).

Why is there a minus sign in the formula?

Because as the angle C grows from 0 to 180 degrees, cosine drops from 1 to -1. The minus sign reflects that side c gets longer as the opposite angle gets larger.

How is the Law of Cosines derived?

Drop an altitude from one vertex to the opposite side. Apply the Pythagorean Theorem to the two resulting right triangles, then combine. Algebraic manipulation produces the Law of Cosines.

Where is the Law of Cosines used?

Surveying, navigation (locating positions from distances), engineering (truss analysis), and any field requiring triangle measurements from incomplete data.

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 covers the unit circle, identities, and applications. Pre-Calculus for Beginners is the natural next step.