Trigonometry Practice

Law of Sines Practice

Practice matching sides with opposite angles, finding missing sides and angles, and checking SSA ambiguous cases.

Start Practicing Review Law of Sines

Try each problem first, then click Show solution to check your side-angle pairing, proportion, algebra, and final answer.

Keep This Relationship Nearby

Law of Sines Essentials

Core Formula

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Opposite Pairs

\[ A\leftrightarrow a \qquad B\leftrightarrow b \qquad C\leftrightarrow c \]

Angle Sum

\[ A+B+C=180^\circ \]
Download Practice PDF Review Law of Sines

Level 1

Match Sides with Opposite Angles

Before solving anything, make sure each side is paired with the angle directly across from it.

Problem 1

Match the Pairs

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

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

Step-by-Step Solution

Side \(a\) is the side directly across from angle \(A\).

\[ A \leftrightarrow a \]
Answer: \[ \boxed{a} \]

Problem 2

Match the Pairs

Which side-angle pair is matched correctly?

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

Step-by-Step Solution

Matching letters belong together:

\[ A\leftrightarrow a, \qquad B\leftrightarrow b, \qquad C\leftrightarrow c \]
Answer: \[ \boxed{ B\leftrightarrow b } \]

Problem 3

Check the Setup

Suppose \(A=40^\circ\), \(a=9\), and \(B=65^\circ\). Which proportion correctly uses the Law of Sines to find \(b\)?

\[ \frac{9}{\sin 40^\circ} = \frac{b}{\sin 65^\circ} \]
\[ \frac{9}{\sin 65^\circ} = \frac{b}{\sin 40^\circ} \]
\[ \frac{40}{\sin 9^\circ} = \frac{65}{\sin b} \]

Step-by-Step Solution

Side \(a=9\) must stay paired with angle \(A=40^\circ\), and side \(b\) must stay paired with angle \(B=65^\circ\).

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
\[ \frac{9}{\sin 40^\circ} = \frac{b}{\sin 65^\circ} \]
Answer: \[ \boxed{ \frac{9}{\sin 40^\circ} = \frac{b}{\sin 65^\circ} } \]

Level 2

Find Missing Sides

Use a known side-angle pair and the pair containing the unknown side.

Problem 4

Find a Missing Side

In triangle \(ABC\), \(A=38^\circ\), \(B=72^\circ\), and \(a=12\). Find \(b\) to the nearest hundredth.

A = 38° B = 72° C a = 12 b = ? c

Step-by-Step Solution

Side \(a=12\) is opposite angle \(A=38^\circ\), and side \(b\) is opposite angle \(B=72^\circ\).

\[ \frac{12}{\sin 38^\circ} = \frac{b}{\sin 72^\circ} \]

Solve for \(b\).

\[ b = \frac{ 12\sin 72^\circ }{ \sin 38^\circ } \]
\[ b \approx 18.54 \]
Answer: \[ \boxed{b\approx18.54} \]

Problem 5

Find a Missing Side

In triangle \(ABC\), \(A=51^\circ\), \(C=64^\circ\), and \(c=15\). Find \(a\) to the nearest hundredth.

A = 51° B C = 64° a = ? b c = 15

Step-by-Step Solution

Side \(c=15\) is opposite angle \(C=64^\circ\), and side \(a\) is opposite angle \(A=51^\circ\).

\[ \frac{a}{\sin 51^\circ} = \frac{15}{\sin 64^\circ} \]

Solve for \(a\).

\[ a = \frac{ 15\sin 51^\circ }{ \sin 64^\circ } \]
\[ a \approx 12.97 \]
Answer: \[ \boxed{a\approx12.97} \]

Problem 6

Find the Third Angle First

In triangle \(ABC\), \(A=44^\circ\), \(B=63^\circ\), and \(a=9\). Find side \(c\) to the nearest hundredth.

A = 44° B = 63° C = ? a = 9 b c = ?

Step-by-Step Solution

We need angle \(C\) before we can pair it with side \(c\).

\[ C = 180^\circ - 44^\circ - 63^\circ \]
\[ C = 73^\circ \]

Now use the complete pair \(a=9\) and \(A=44^\circ\).

\[ \frac{9}{\sin 44^\circ} = \frac{c}{\sin 73^\circ} \]

Solve for \(c\).

\[ c = \frac{ 9\sin 73^\circ }{ \sin 44^\circ } \]
\[ c \approx 12.39 \]
Answer: \[ \boxed{c\approx12.39} \]

Level 3

Find Missing Angles

Use the Law of Sines to isolate a sine value, then use inverse sine to find the missing angle.

Calculator reminder: Use degree mode when evaluating inverse sine.

Problem 7

Find a Missing Angle

In triangle \(ABC\), \(A=35^\circ\), \(a=10\), and \(b=14\). Find angle \(B\) to the nearest tenth.

A = 35° B = ? C a = 10 b = 14 c

Step-by-Step Solution

Use the known pair \(a=10\) and \(A=35^\circ\), then pair side \(b=14\) with angle \(B\).

\[ \frac{10}{\sin 35^\circ} = \frac{14}{\sin B} \]

Solve for \(\sin B\).

\[ \sin B = \frac{ 14\sin 35^\circ }{ 10 } \]

Use inverse sine.

\[ B = \sin^{-1} \left( \frac{ 14\sin 35^\circ }{ 10 } \right) \]
\[ B \approx 53.4^\circ \]
First Answer: \[ \boxed{ B\approx53.4^\circ } \]

Problem 8

Find a Missing Angle

In triangle \(ABC\), \(C=48^\circ\), \(c=16\), and \(a=11\). Find angle \(A\) to the nearest tenth.

A = ? B C = 48° a = 11 b c = 16

Step-by-Step Solution

Use the known pair \(c=16\) and \(C=48^\circ\).

\[ \frac{11}{\sin A} = \frac{16}{\sin 48^\circ} \]

Solve for \(\sin A\).

\[ \sin A = \frac{ 11\sin 48^\circ }{ 16 } \]
\[ A = \sin^{-1} \left( \frac{ 11\sin 48^\circ }{ 16 } \right) \]
\[ A \approx 30.7^\circ \]
Answer: \[ \boxed{ A\approx30.7^\circ } \]

Problem 9

SSA Check

In triangle \(ABC\), \(A=36^\circ\), \(a=8\), and \(b=11\). Find the first possible value of angle \(B\).

A = 36° B = ? C a = 8 b = 11 c

Step-by-Step Solution

Use the known pair \(a=8\) and \(A=36^\circ\).

\[ \frac{8}{\sin 36^\circ} = \frac{11}{\sin B} \]

Solve for \(\sin B\).

\[ \sin B = \frac{ 11\sin 36^\circ }{ 8 } \]
\[ B = \sin^{-1} \left( \frac{ 11\sin 36^\circ }{ 8 } \right) \]
\[ B_1 \approx 53.9^\circ \]
First Possible Angle: \[ \boxed{ B_1\approx53.9^\circ } \]
Do not stop here. Because this is an SSA problem, we still need to check the supplementary angle.

Level 4

The SSA Ambiguous Case

Check whether the supplementary angle creates a second valid triangle.

Problem 10

Two Triangles

In triangle \(ABC\), \(A=36^\circ\), \(a=8\), and \(b=11\). Determine all possible values of angle \(B\).

Step-by-Step Solution

From Problem 9, the first possible angle is:

\[ B_1 \approx 53.9^\circ \]

Now find the supplementary angle.

\[ B_2 = 180^\circ - 53.9^\circ \]
\[ B_2 \approx 126.1^\circ \]

Check whether the second angle is valid with \(A=36^\circ\).

\[ 36^\circ + 126.1^\circ = 162.1^\circ \]

Since this is less than \(180^\circ\), a positive third angle still remains.

Answers: \[ \boxed{ B\approx53.9^\circ \text{ or } 126.1^\circ } \]

Problem 11

One Triangle

In triangle \(ABC\), \(A=50^\circ\), \(a=14\), and \(b=10\). Determine how many triangles are possible and find angle \(B\).

Step-by-Step Solution

Use the Law of Sines.

\[ \frac{14} {\sin 50^\circ} = \frac{10} {\sin B} \]

Solve for \(\sin B\).

\[ \sin B = \frac{ 10\sin 50^\circ }{ 14 } \]
\[ B_1 \approx 33.2^\circ \]

Now check the supplement.

\[ B_2 = 180^\circ - 33.2^\circ = 146.8^\circ \]

Test it with \(A=50^\circ\).

\[ 50^\circ + 146.8^\circ = 196.8^\circ \]

That exceeds \(180^\circ\), so the supplementary angle cannot form a triangle.

Answer: \[ \boxed{ \text{One triangle, } B\approx33.2^\circ } \]

Problem 12

No Triangle

In triangle \(ABC\), \(A=30^\circ\), \(a=4\), and \(b=10\). Determine whether a triangle is possible.

Step-by-Step Solution

Use the Law of Sines.

\[ \frac{4} {\sin 30^\circ} = \frac{10} {\sin B} \]

Solve for \(\sin B\).

\[ \sin B = \frac{ 10\sin 30^\circ }{ 4 } \]
\[ \sin B = \frac{5}{4} = 1.25 \]

But the sine of a real angle cannot be greater than \(1\).

Answer: \[ \boxed{ \text{No triangle} } \]

Before You Finish

Law of Sines Checklist

  1. Match each side with its opposite angle.
  2. Find at least one complete side-angle pair.
  3. Write two matching Law of Sines ratios.
  4. Solve with algebra or inverse sine.
  5. If the problem is SSA, check the supplementary angle.

Continue Learning

Related Law of Sines Resources

Review the lesson, strengthen your right-triangle trig skills, or continue into the Law of Cosines.

Law of Sines Guide

Review side-angle pairs, missing sides and angles, and the SSA ambiguous case.

SOH-CAH-TOA

Review sine, cosine, and tangent for solving right triangles.

SOH-CAH-TOA Practice

Practice missing sides and angles before working with oblique triangles.

Special Right Triangles

Review exact side relationships in 45-45-90 and 30-60-90 triangles.

Radians vs Degrees

Review angle measure and common degree-radian conversions.

Trigonometry Resources

Browse more trig guides, practice pages, worksheets, and learning tools.

Personalized Math Support

Need Help Making Sense of the Math?

If a concept still feels confusing, you do not have to work through it alone. Tell me what you are studying and where you are getting stuck, and I will help you determine the best next step.

Contact Raul

Support Free Math Resources

Find this resource helpful?

RaulTheTutor.com provides free interactive tools, guides, reference resources, and practice materials for students, parents, and teachers. If this resource helped you, consider making an optional contribution to support the development of more free math resources.

Support Free Math Resources