When a triangle does not contain a right angle, the familiar Pythagorean theorem and basic right-triangle trigonometry are not enough to solve every problem. This is where the Law of Sines and Law of Cosines become essential. These two powerful trigonometric laws allow us to find missing sides and angles in non-right triangles, making them fundamental tools in geometry, trigonometry, surveying, navigation, engineering, physics, and many real-world applications.
A strong understanding of these laws helps us solve triangles even when the information provided does not include a 90-degree angle. By identifying the known sides and angles correctly, choosing the appropriate law, and organizing our calculations carefully, we can solve a wide variety of complex-looking triangle problems.
What Is a Non-Right Triangle?
A non-right triangle is any triangle that does not contain a 90-degree angle. There are two main types:
Acute triangles — all three angles are less than 90°.
Obtuse triangles — one angle is greater than 90°.
Every triangle has three sides and three angles. Traditionally, we label the angles as A, B, and C, while the sides opposite those angles are labeled a, b, and c, respectively.
This relationship between sides and opposite angles is extremely important when using the Law of Sines.
For example:
Side a is opposite angle A.
Side b is opposite angle B.
Side c is opposite angle C.
The three angles of every triangle also satisfy:
[
A+B+C=180^\circ
]
This basic fact becomes useful when calculating the final unknown angle.
What Is the Law of Sines?
The Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles.
The formula is:
[
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
]
We can also write it as:
[
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}
]
Both forms express the same relationship.
The Law of Sines is especially useful when we know one side and its opposite angle, along with another side or angle.
When Should We Use the Law of Sines?
We generally use the Law of Sines when the given information matches one of these situations:
1. ASA — Angle-Side-Angle
We know two angles and the side between them.
2. AAS — Angle-Angle-Side
We know two angles and a side that is not between them.
3. SSA — Side-Side-Angle
We know two sides and an angle opposite one of those sides.
The SSA case requires special care because it can sometimes produce two possible triangles, one triangle, or no triangle at all. This situation is commonly called the ambiguous case.
Law of Sines Example: Finding a Missing Side
Suppose we have a triangle with:
[A=40^\circ]
[B=65^\circ]
and
[a=10]
We want to find side (b).
Using the Law of Sines:
[\frac{a}{\sin A}=\frac{b}{\sin B}]
Substitute the known values:
[\frac{10}{\sin40^\circ}=\frac{b}{\sin65^\circ}]
Solving for (b):
[b=\frac{10\sin65^\circ}{\sin40^\circ}]
Therefore:
[b\approx14.09]
So the missing side is approximately:
[\boxed{14.09}]
Finding a Missing Angle Using the Law of Sines
The Law of Sines can also help us find unknown angles.
Suppose:
[a=12]
[A=35^\circ]
and:
[b=15]
We can use:
[\frac{\sin B}{b}=\frac{\sin A}{a}]
Therefore:
[\sin B=\frac{15\sin35^\circ}{12}]
After calculating:
[\sin B\approx0.717]
We then use the inverse sine function:
[B=\sin^{-1}(0.717)]
which gives approximately:
[B\approx45.8^\circ]
We must always check whether a second possible angle exists, particularly when solving an SSA problem
The Ambiguous Case of the Law of Sines
One of the most important concepts associated with the Law of Sines is the ambiguous case.
When we are given two sides and an angle that is not included between them, the information may allow:
No triangle
One triangle
Two different triangles
This happens because the sine function has the same positive value for two supplementary angles:
[\sin\theta=\sin(180^\circ-\theta)]
For example, if:
[\sin B=0.6]
then:
[B\approx36.87^\circ]
But another possible angle is:
[180^\circ-36.87^\circ=143.13^\circ]
We must check whether both angles can fit into a triangle with the other known angle.
This is why we should never automatically accept the first answer produced by the inverse sine function.
What Is the Law of Cosines?
The Law of Cosines extends the Pythagorean theorem to all types of triangles.
Its standard forms are:
[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]
The Law of Cosines is particularly useful when we know two sides and the included angle, or when we know all three sides and need to determine an angle.
When Should We Use the Law of Cosines?
We generally choose the Law of Cosines in two major situations.
SAS — Side-Angle-Side
We know two sides and the angle between them.
For example:
[b=8,\quad c=11,\quad A=60^\circ]
We can find side (a) using:
[a^2=b^2+c^2-2bc\cos A]
SSS — Side-Side-Side
We know all three sides and need to find one or more angles.
For example:
[a=7,\quad b=8,\quad c=10]
We can rearrange the Law of Cosines to find angle (A):
[\cos A=\frac{b^2+c^2-a^2}{2bc}]
Then:
[A=\cos^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)]
Law of Cosines Example: Finding a Missing Side
Suppose two sides of a triangle measure:
[b=7]
and
[c=10]
and the included angle is:
[A=60^\circ]
We want to find side (a).
Start with:
[a^2=b^2+c^2-2bc\cos A]
Substitute:
[a^2=7^2+10^2-2(7)(10)\cos60^\circ]
Since:
[\cos60^\circ=\frac12]
we get:
[a^2=49+100-140\left(\frac12\right)]
[a^2=149-70]
[a^2=79]
Therefore:
[a=\sqrt{79}]
and:
[\boxed{a\approx8.89}]
Law of Cosines Example: Finding an Angle
Suppose all three sides of a triangle are known:
[a=8,\quad b=10,\quad c=13]
We want to find angle (A).
Starting with:
[a^2=b^2+c^2-2bc\cos A]
we rearrange:
[2bc\cos A=b^2+c^2-a^2]
Therefore:
[\cos A=\frac{b^2+c^2-a^2}{2bc}]
Substituting:
[\cos A=\frac{10^2+13^2-8^2}{2(10)(13)}]
[\cos A=\frac{100+169-64}{260}]
[\cos A=\frac{205}{260}]
[\cos A\approx0.7885]
Therefore:
[A=\cos^{-1}(0.7885)]
So:
[\boxed{A\approx37.9^\circ}]
Law of Sines vs. Law of Cosines
Choosing the correct law is often the most important step.
| Given Information | Best Method |
|---|---|
| ASA | Law of Sines |
| AAS | Law of Sines |
| SSA | Law of Sines |
| SAS | Law of Cosines |
| SSS | Law of Cosines |
A simple strategy is to look for an opposite side-angle pair.
If we already know a side and its opposite angle, the Law of Sines is often the natural choice.
If we know two sides and the included angle, the Law of Cosines is generally the better option.
If all three sides are known, we must use the Law of Cosines to determine an angle.
How to Solve a Non-Right Triangle Step by Step
We can use the following process for almost any problem involving the Law of Sines or Law of Cosines.
Step 1: Label the Triangle
Identify:
[
A,\ B,\ C
]
and their opposite sides:
[
a,\ b,\ c
]
Step 2: List the Given Information
Write down every known side and angle.
Step 3: Identify the Triangle Case
Determine whether the problem is:
ASA
AAS
SSA
SAS
SSS
Step 4: Choose the Appropriate Law
Use the Law of Sines for ASA, AAS, and many SSA problems.
Use the Law of Cosines for SAS and SSS problems.
Step 5: Calculate the Unknown
Substitute the known values carefully and solve.
Step 6: Use the Triangle Angle Sum
Remember:
[A+B+C=180^\circ]
If two angles are known, the third is:
[C=180^\circ-A-B]
Step 7: Check the Answer
Make sure:
All angles are positive.
The angles add to 180°.
The largest angle is opposite the longest side.
The smallest angle is opposite the shortest side.
These checks can quickly reveal calculation or setup errors.
How the Law of Cosines Relates to the Pythagorean Theorem
The Law of Cosines becomes the Pythagorean theorem when the included angle is 90°.
Starting with:
[c^2=a^2+b^2-2ab\cos C]
If:
[C=90^\circ]
then:
[\cos90^\circ=0]
Therefore:
[c^2=a^2+b^2]
This is exactly the Pythagorean theorem.
This connection shows that the Law of Cosines is a broader rule that works for right, acute, and obtuse triangles.
Solving Real-World Problems With Non-Right Triangles
The Law of Sines and Law of Cosines are not limited to textbook exercises. We can apply them to many practical situations.
Surveying and Land Measurement
Surveyors can use known distances and angles to calculate distances that are difficult to measure directly.
Navigation
Triangles can help determine distances between locations when directions and known distances are available.
Construction and Engineering
Engineers can use triangle relationships to determine lengths, angles, slopes, and structural dimensions.
Astronomy
Triangulation techniques can use angles and distances to estimate the positions of distant objects.
Physics
Vector problems frequently create triangles in which the Law of Sines and Law of Cosines can help determine magnitudes and directions.
Common Mistakes With the Law of Sines and Law of Cosines
Even when the formulas are understood, several mistakes occur frequently.
Mistake 1: Matching the Wrong Side and Angle
In the Law of Sines, each side must correspond to its opposite angle.
The correct pairings are:
[a\leftrightarrow A]
[b\leftrightarrow B]
[c\leftrightarrow C]
Mistake 2: Using the Wrong Law
Using the Law of Sines for an SAS problem can create unnecessary complications. We should identify the given information before choosing a formula.
Mistake 3: Forgetting the Second SSA Solution
The SSA case can produce two possible triangles. We should always investigate whether the supplementary angle also works.
Mistake 4: Calculator in the Wrong Mode
When working with angles measured in degrees, the calculator should be set to degree mode.
Mistake 5: Rounding Too Early
Keeping several decimal places throughout the calculation gives us a more accurate final result.
Tips for Solving Law of Sines and Law of Cosines Problems Faster
We can improve both speed and accuracy by following a few habits.
First, draw the triangle. Even a simple sketch makes the relationships between sides and angles much easier to see.
Second, label opposite pairs clearly. This prevents one of the most common Law of Sines mistakes.
Third, identify the case before calculating. Knowing whether the problem is SSS, SAS, ASA, AAS, or SSA immediately narrows down the correct method.
Fourth, keep exact values when possible. Avoid excessive rounding until the final answer.
Finally, check whether the answer makes geometric sense. A longer side should be opposite a larger angle, while a shorter side should be opposite a smaller angle.
Quick Reference: Law of Sines and Law of Cosines
Law of Sines
[
\boxed{\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}}
]
Best for:
ASA
AAS
SSA
Law of Cosines
[\boxed{a^2=b^2+c^2-2bc\cos A}]
[\boxed{b^2=a^2+c^2-2ac\cos B}]
[\boxed{c^2=a^2+b^2-2ab\cos C}]
Best for:
SAS
SSS
Final Thoughts
The Law of Sines and Law of Cosines give us reliable methods for solving triangles that cannot be handled using basic right-triangle techniques alone. The Law of Sines connects opposite sides and angles, while the Law of Cosines connects the lengths of three sides with an included angle.
The key to solving these problems successfully is not simply memorizing formulas. We need to recognize the information given, identify the correct triangle case, choose the appropriate law, perform the calculations carefully, and verify that the result makes geometric sense.
Once these patterns become familiar, non-right triangle problems become much more manageable. Whether we are studying trigonometry, preparing for a math exam, working with geometry, or applying mathematics to real-world measurement, the Law of Sines and Law of Cosines provide two of the most useful tools for solving triangles.
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