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

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Math Grade 10-12

Law of Sines Worked Examples Cheat Sheet

A printable reference covering the Law of Sines, triangle labeling, missing sides, missing angles, and ambiguous cases for grades 10-12.

TopicsTrigonometryGeometryMeasurementWord ProblemsMathematicsStudy Guide
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The Law of Sines helps students solve non-right triangles when a pair of matching angle and side measures is known. This cheat sheet focuses on worked-example thinking, including how to set up ratios, choose the correct equation, and check whether the answer makes sense. It is especially useful for cases involving ASAASA, AASAAS, and SSASSA information.

Key Facts

  • The Law of Sines is asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}, where each side is opposite its matching angle.
  • An equivalent form is sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}, which is often convenient when solving for an angle.
  • Use aa opposite AA, bb opposite BB, and cc opposite CC to keep every ratio correctly matched.
  • The angles in any triangle satisfy A+B+C=180∘A+B+C=180^{\circ}.
  • To find a missing side, set up a proportion such as asin⁡A=bsin⁡B\frac{a}{\sin A}=\frac{b}{\sin B} and solve for the unknown side.
  • To find a missing angle, use a proportion such as sin⁡Aa=sin⁡Bb\frac{\sin A}{a}=\frac{\sin B}{b}, then apply A=sin⁡−1(asin⁡Bb)A=\sin^{-1}\left(\frac{a\sin B}{b}\right) when solving for AA.
  • The ambiguous case can occur with SSASSA information because sin⁡θ=sin⁡(180∘−θ)\sin \theta=\sin(180^{\circ}-\theta) may give two possible angles.
  • A triangle is impossible if the computed third angle is 0∘0^{\circ} or less, or if the side and angle relationships contradict the Law of Sines.

Vocabulary

Law of Sines
A rule that relates each side of a triangle to the sine of its opposite angle using asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.
Opposite side
The side across from a given angle, such as side aa being opposite angle AA.
Included angle
An angle located between two known sides of a triangle.
SSA case
A triangle information pattern with two sides and a non-included angle, which may produce zero, one, or two possible triangles.
Ambiguous case
A situation in the SSASSA case where two different triangles can satisfy the same given measurements.
Inverse sine
The operation sin⁡−1(x)\sin^{-1}(x) used to find an angle whose sine value is xx.

Common Mistakes to Avoid

  • Matching a side with the wrong angle is wrong because the Law of Sines only works with opposite pairs such as asin⁡A\frac{a}{\sin A}.
  • Using the Law of Sines without a known opposite pair is wrong because a proportion needs at least one complete side-angle pair.
  • Forgetting the second possible angle in an SSASSA problem is wrong because sin⁡θ=sin⁡(180∘−θ)\sin \theta=\sin(180^{\circ}-\theta) can create two valid triangles.
  • Rounding too early is wrong because intermediate rounding can change the final side length or angle measure noticeably.
  • Accepting an angle sum greater than 180∘180^{\circ} is wrong because every triangle must satisfy A+B+C=180∘A+B+C=180^{\circ}.

Practice Questions

  1. 1 In △ABC\triangle ABC, A=42∘A=42^{\circ}, B=68∘B=68^{\circ}, and a=12a=12. Find bb using the Law of Sines.
  2. 2 In △ABC\triangle ABC, A=35∘A=35^{\circ}, a=9a=9, and b=13b=13. Find the possible value or values of BB.
  3. 3 In △ABC\triangle ABC, B=51∘B=51^{\circ}, C=74∘C=74^{\circ}, and c=18c=18. Find AA and then find bb.
  4. 4 Explain why an SSASSA problem can have two possible triangles, but an ASAASA problem has only one triangle.

Understanding Law of Sines Worked Examples

The most important habit is to draw a clean triangle before doing any calculator work. Put each capital letter at a corner and write its matching lowercase side across from it. This prevents the most common error, which is pairing a side with the angle next to it.

Mark the information you know directly on the sketch. If a side is longer than another side, its opposite angle must be larger. This simple visual check catches many wrong answers before they spread through a calculation.

When you solve for a side, keep extra decimal places until the final step. Early rounding can noticeably change a later angle, especially in a narrow triangle. When you solve for an angle, make sure the calculator is set to degree mode.

The inverse sine button gives one angle between zero degrees and ninety degrees for a positive result. That output may be the correct angle, but it may be only one of two possible angles. Write down the calculator value first, then examine the triangle rather than accepting it automatically.

The two-answer situation has a geometric cause. A known side can sometimes swing into two positions while still reaching a point that creates the same sine value. One position makes an acute angle.

The other makes an obtuse angle whose measure is found by subtracting the acute result from one hundred eighty degrees. Each candidate must leave a positive amount for the final angle. It must fit the given side lengths too.

An obtuse angle must face the longest side, so reject it if a longer known side would face a smaller angle. Sometimes neither position works, and sometimes only one works.

These ideas appear whenever indirect measurements form a triangle. Surveyors can estimate a distance across a river from two observation points. Engineers use triangular frames because triangles hold their shape under load.

Navigation and mapping use measured directions and distances to locate positions. In school problems, the labels may represent roads, cables, roof braces, or lines of sight, but the reasoning stays the same. A strong final check compares the size order of all sides with the size order of their opposite angles.

Then add the three angles mentally and ask whether the shape looks possible. Calculations provide numbers, but the diagram and these checks tell you whether those numbers describe a real triangle.