Trigonometry Practice

Law of Cosines Practice

Practice choosing the correct formula, solving SAS and SSS triangles, and finding missing sides and angles.

Start Practicing Review Law of Cosines

Try each problem first, then click Show solution to check your formula choice, substitution, algebra, and final answer.

Keep These Relationships Nearby

Law of Cosines Essentials

Find Side \(a\)

\[ a^2 = b^2+c^2 - 2bc\cos A \]

Find Side \(b\)

\[ b^2 = a^2+c^2 - 2ac\cos B \]

Find Side \(c\)

\[ c^2 = a^2+b^2 - 2ab\cos C \]
Download Practice PDF Review Law of Cosines

Level 1

Choose the Correct Formula

Match the side you are finding with the angle directly opposite it.

Problem 1

Match the Opposite Pair

In triangle \(ABC\), which angle is opposite side \(c\)?

A B C a b c
\(A\)
\(B\)
\(C\)

Step-by-Step Solution

Side \(c\) is directly opposite angle \(C\).

\[ c \leftrightarrow C \]
Answer: \[ \boxed{C} \]

Problem 2

Choose the Formula

Which formula should you use if you are trying to find side \(b\)?

\[ a^2 = b^2+c^2 - 2bc\cos A \]
\[ b^2 = a^2+c^2 - 2ac\cos B \]
\[ c^2 = a^2+b^2 - 2ab\cos C \]

Step-by-Step Solution

The side being found is \(b\), so use the version of the Law of Cosines with \(b^2\) on the left.

\[ b^2 = a^2+c^2 - 2ac\cos B \]
Answer: \[ \boxed{ b^2 = a^2+c^2 - 2ac\cos B } \]

Problem 3

Choose the Method

A triangle has two known sides and the included angle between them. Which method should you use first?

SOH-CAH-TOA
Law of Sines
Law of Cosines

Step-by-Step Solution

Two sides and the included angle form an SAS case.

SAS is one of the main cases where the Law of Cosines is the natural first method.

Answer: Law of Cosines

Level 2

Find Missing Sides

Use SAS information to solve for the side opposite the known included angle.

Problem 4

Find a Missing Side

In triangle \(ABC\), \(a=8\), \(b=11\), and \(C=47^\circ\). Find side \(c\) to the nearest hundredth.

A B C = 47° a = 8 b = 11 c = ?

Step-by-Step Solution

Since side \(c\) is opposite angle \(C\), use:

\[ c^2 = a^2+b^2 - 2ab\cos C \]

Substitute the known values.

\[ c^2 = 8^2+11^2 - 2(8)(11)\cos(47^\circ) \]
\[ c^2 \approx 64.97 \]

Take the square root.

\[ c \approx \sqrt{64.97} \]
Answer: \[ \boxed{ c\approx8.06 } \]

Problem 5

Find a Missing Side

In triangle \(ABC\), \(a=13\), \(c=9\), and \(B=58^\circ\). Find side \(b\) to the nearest hundredth.

A B = 58° C a = 13 b = ? c = 9

Step-by-Step Solution

Since side \(b\) is opposite angle \(B\), use:

\[ b^2 = a^2+c^2 - 2ac\cos B \]

Substitute.

\[ b^2 = 13^2+9^2 - 2(13)(9)\cos(58^\circ) \]
\[ b^2 \approx 126.00 \]

Take the square root.

\[ b \approx \sqrt{126.00} \]
Answer: \[ \boxed{ b\approx11.22 } \]

Problem 6

Find a Missing Side

In triangle \(ABC\), \(b=7\), \(c=15\), and \(A=102^\circ\). Find side \(a\) to the nearest hundredth.

A = 102° B C a = ? b = 7 c = 15

Step-by-Step Solution

Since side \(a\) is opposite angle \(A\), use:

\[ a^2 = b^2+c^2 - 2bc\cos A \]

Substitute.

\[ a^2 = 7^2+15^2 - 2(7)(15)\cos(102^\circ) \]
\[ a^2 \approx 317.66 \]

Take the square root.

\[ a \approx \sqrt{317.66} \]
Answer: \[ \boxed{ a\approx17.82 } \]

Level 3

Find Missing Angles

Use SSS information to isolate cosine, then use inverse cosine to recover the missing angle.

Calculator reminder: Use degree mode when evaluating inverse cosine.

Problem 7

Find a Missing Angle

In triangle \(ABC\), \(a=7\), \(b=10\), and \(c=12\). Find angle \(C\) to the nearest tenth.

A B C = ? a = 7 b = 10 c = 12

Step-by-Step Solution

Since we are finding angle \(C\), start with:

\[ c^2 = a^2+b^2 - 2ab\cos C \]

Substitute the three side lengths.

\[ 12^2 = 7^2+10^2 - 2(7)(10)\cos C \]
\[ 144 = 149 - 140\cos C \]

Solve for \(\cos C\).

\[ 140\cos C = 5 \] \[ \cos C = \frac{1}{28} \]

Use inverse cosine.

\[ C = \cos^{-1} \left( \frac{1}{28} \right) \]
Answer: \[ \boxed{ C\approx88.0^\circ } \]

Problem 8

Find a Missing Angle

In triangle \(ABC\), \(a=9\), \(b=14\), and \(c=17\). Find angle \(A\) to the nearest tenth.

A = ? B C a = 9 b = 14 c = 17

Step-by-Step Solution

Since we are finding angle \(A\), use the formula with \(a^2\) and \(\cos A\).

\[ a^2 = b^2+c^2 - 2bc\cos A \]

Substitute.

\[ 9^2 = 14^2+17^2 - 2(14)(17)\cos A \]
\[ 81 = 485 - 476\cos A \]

Solve for \(\cos A\).

\[ 476\cos A = 404 \] \[ \cos A = \frac{404}{476} \]

Use inverse cosine.

\[ A = \cos^{-1} \left( \frac{404}{476} \right) \]
Answer: \[ \boxed{ A\approx32.0^\circ } \]

Problem 9

Find a Missing Angle

In triangle \(ABC\), \(a=11\), \(b=13\), and \(c=18\). Find angle \(B\) to the nearest tenth.

A B = ? C a = 11 b = 13 c = 18

Step-by-Step Solution

Since we are finding angle \(B\), use:

\[ b^2 = a^2+c^2 - 2ac\cos B \]

Substitute.

\[ 13^2 = 11^2+18^2 - 2(11)(18)\cos B \]
\[ 169 = 445 - 396\cos B \]

Solve for \(\cos B\).

\[ 396\cos B = 276 \] \[ \cos B = \frac{276}{396} \]

Use inverse cosine.

\[ B = \cos^{-1} \left( \frac{276}{396} \right) \]
Answer: \[ \boxed{ B\approx45.8^\circ } \]

Level 4

Mixed Law of Cosines Problems

Decide what information you have, choose the correct form of the Law of Cosines, and solve.

Problem 10

Mixed Practice

In triangle \(ABC\), \(a=12\), \(b=16\), and \(C=73^\circ\). Find side \(c\) to the nearest hundredth.

A B C = 73° a = 12 b = 16 c = ?

Step-by-Step Solution

We know two sides and the included angle, so this is an SAS problem.

Since \(c\) is opposite angle \(C\), use:

\[ c^2 = a^2+b^2 - 2ab\cos C \]

Substitute.

\[ c^2 = 12^2+16^2 - 2(12)(16)\cos(73^\circ) \]
\[ c^2 \approx 287.73 \]

Take the square root.

\[ c \approx \sqrt{287.73} \]
Answer: \[ \boxed{ c\approx16.96 } \]

Problem 11

Mixed Practice

In triangle \(ABC\), \(a=15\), \(b=18\), and \(c=20\). Find angle \(A\) to the nearest tenth.

A = ? B C a = 15 b = 18 c = 20

Step-by-Step Solution

We know all three sides, so this is an SSS problem.

Since we are finding angle \(A\), use:

\[ a^2 = b^2+c^2 - 2bc\cos A \]

Substitute.

\[ 15^2 = 18^2+20^2 - 2(18)(20)\cos A \]
\[ 225 = 724 - 720\cos A \]

Solve for \(\cos A\).

\[ 720\cos A = 499 \] \[ \cos A = \frac{499}{720} \]

Use inverse cosine.

\[ A = \cos^{-1} \left( \frac{499}{720} \right) \]
Answer: \[ \boxed{ A\approx46.1^\circ } \]

Problem 12

Method Selection

A triangle has side lengths \(9\), \(12\), and \(14\). You want to find its largest angle. Which angle should you find first, and why?

Think first: The largest angle of a triangle is always opposite its longest side.

Step-by-Step Solution

The longest side has length \(14\).

Therefore, the largest angle must be the angle directly opposite the side of length \(14\).

If we call that side \(c\), then the angle opposite it is angle \(C\).

\[ c=14 \quad\Longrightarrow\quad \text{find }C \]

Since all three sides are known, use the Law of Cosines:

\[ c^2 = a^2+b^2 - 2ab\cos C \]
Answer: Find the angle opposite the side of length 14.

Before You Finish

Law of Cosines Checklist

  1. Identify whether the problem is SAS or SSS.
  2. Match each side with the angle directly opposite it.
  3. Choose the Law of Cosines formula that matches the side or angle you are finding.
  4. Substitute carefully and keep the cosine term with the correct angle.
  5. Take a square root when finding a side.
  6. Use inverse cosine when finding an angle.
  7. Round only the final answer.

Continue Learning

Related Law of Cosines Resources

Review the lesson, compare the Law of Sines and Law of Cosines, or continue strengthening your triangle-solving skills.

Law of Cosines Guide

Review SAS, SSS, missing sides, missing angles, and formula selection.

Law of Sines

Review ASA, AAS, SSA, and the ambiguous case.

Law of Sines Practice

Practice side-angle pairing, missing sides, missing angles, and SSA cases.

SOH-CAH-TOA

Review sine, cosine, and tangent for right triangles.

Special Right Triangles

Review 45-45-90 and 30-60-90 triangle relationships.

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