How to Use the Law of Sines to Find Sides and Angles in a Triangle
The concept of Law of Sines plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding this formula is crucial for solving many triangle-related problems, especially when you cannot use the basic SOH-CAH-TOA rules or Pythagoras’ theorem.
What Is Law of Sines?
The Law of Sines is defined as a rule that relates the sides and angles of any triangle (not just right-angled triangles). You’ll find this concept applied in areas such as non-right triangles in trigonometry, navigation, and triangle problem-solving in physics and engineering.
Key Formula for Law of Sines
Here’s the standard formula:
\( \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \)
where a, b, c are side lengths of the triangle, and A, B, C are their opposite angles.
Cross-Disciplinary Usage
Law of Sines is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various trigonometry and geometric reasoning questions.
Step-by-Step Illustration
- Suppose you know two angles (A, B) and side a in a triangle.
Given: \( a = 7 \) cm, \( \angle A = 40^\circ \), \( \angle B = 65^\circ \) - Find angle C.
\( \angle C = 180^\circ - 40^\circ - 65^\circ = 75^\circ \) - Apply Law of Sines to find side b.
\( \dfrac{a}{\sin A} = \dfrac{b}{\sin B} \)
\( b = \dfrac{a \cdot \sin B}{\sin A} \)
\( b = \dfrac{7 \cdot \sin 65^\circ}{\sin 40^\circ} \)
\( b \approx \dfrac{7 \cdot 0.9063}{0.6428} \approx 9.87 \) cm
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with Law of Sines—especially the ambiguous SSA case. Many students use this trick during timed exams to quickly know if 0, 1, or 2 triangles are possible:
- To check if you have 0/1/2 solutions:
If \( a < b \), and \( \angle A \) is acute:
a. If \( a < b \sin A \) — No solution
b. If \( a = b \sin A \) — One right triangle
c. If \( b \sin A < a < b \) — Two triangles possible
d. If \( a \geq b \) — One triangle
Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE and JEE. Vedantu’s Trigonometry resources include such shortcuts to help you build speed and accuracy.
Try These Yourself
- Solve for side c in a triangle where \( a = 8 \), \( A = 45^\circ \), \( C = 75^\circ \), and \( c = ? \)
- Check if the following triangle is possible using Law of Sines: \( a = 3 \), \( b = 7 \), \( A = 30^\circ \)
- Find angle B if \( a = 10 \), \( b = 14 \), \( A = 40^\circ \)
- In triangle PQR, if \( PQ = 6 \), \( PR = 5 \), and \( \angle Q = 50^\circ \), find \( \angle R \) using Law of Sines.
Frequent Errors and Misunderstandings
- Mixing up when to use Law of Sines and Law of Cosines (use Sines for ASA, AAS, SSA cases, not for SSS/SAS only).
- Forgetting to check for the "ambiguous case" (SSA) which might give two possible triangles or no solution.
- Making sine inverse errors: forgetting that \( \sin^{-1}(x) \) can have two possible angles between 0° and 180°.
- Not converting angle measures correctly (degrees vs radians) in calculations.
Relation to Other Concepts
The idea of Law of Sines connects closely with topics such as Law of Cosines and core Trigonometry. Mastering this helps with understanding more advanced triangle properties and geometric problem-solving.
Classroom Tip
A quick way to remember Law of Sines is: “A side divided by the sine of its opposite angle is the same for all sides.” Draw a triangle, label sides and angles, and visually pair each side to its opposite angle. Vedantu’s teachers often use color-coding during live classes for this pairing.
We explored Law of Sines—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.
Related Pages: Law of Cosines, Sin, Cos, Tan Values, Types of Triangles
FAQs on Law of Sines: Formula, Solved Examples & Ambiguous Case
1. What is the Law of Sines and when is it used?
The Law of Sines is a fundamental rule in trigonometry that states the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides. It is used to find unknown side lengths or angle measures in any triangle (not just right-angled ones) when you know certain combinations of sides and angles, specifically for cases like Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), and the special Side-Side-Angle (SSA) case.
2. What is the formula for the Law of Sines?
For a triangle with sides a, b, and c, and opposite angles A, B, and C respectively, the Law of Sines formula is expressed as:
a/sin(A) = b/sin(B) = c/sin(C)
This means you can set any two ratios equal to each other to solve for a missing side or angle, provided you have at least one complete side-angle pair.
3. When should you use the Law of Sines instead of the Law of Cosines?
You should choose the appropriate law based on the given information about the triangle.
- Use the Law of Sines when you have a known side-angle pair. This applies to scenarios like Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), or the Side-Side-Angle (SSA) case.
- Use the Law of Cosines when you do not have a known side-angle pair. This applies to scenarios where you know Side-Angle-Side (SAS) or Side-Side-Side (SSS).
4. What is the 'ambiguous case' in the Law of Sines?
The ambiguous case, known as the Side-Side-Angle (SSA) case, occurs when we are given two sides and a non-included angle. This situation is 'ambiguous' because the given information might result in zero, one, or two possible triangles. The ambiguity arises because the sine of an angle and the sine of its supplement (180° - angle) are equal, potentially allowing for two valid solutions for an angle.
5. How do you determine if the ambiguous case (SSA) results in zero, one, or two triangles?
To determine the number of possible triangles in the SSA case (given sides 'a' and 'b' and angle 'A'), you can compare side 'a' to the triangle's height (h = b * sin A).
- If a < h, the side is too short to form a triangle. Zero solutions.
- If a = h, the side forms a right-angled triangle. One solution.
- If h < a < b, the side can swing to form two different triangles (one acute, one obtuse). Two solutions.
- If a ≥ b, the side is too long to swing back and form a second triangle. One solution.
6. What is the most common mistake students make when applying the Law of Sines?
A frequent error is forgetting to check for the ambiguous case (SSA). When solving for an angle using the inverse sine function (sin⁻¹), a calculator typically provides only the acute angle solution. Students often forget that a second, obtuse angle (180° - acute angle) might also be a valid solution, leading to a second possible triangle. Always check if this second angle is possible by ensuring the sum of all angles does not exceed 180°.
7. Can the Law of Sines be applied to right-angled triangles?
Yes, the Law of Sines can be applied to right-angled triangles, but it's often an unnecessarily complex method. For right-angled triangles, the basic trigonometric ratios (SOH-CAH-TOA) and the Pythagorean theorem are much more direct and efficient for solving for unknown sides and angles. The primary power of the Law of Sines is its ability to solve non-right (oblique) triangles.
8. What are some real-world applications of the Law of Sines?
The Law of Sines is crucial in fields that require precise distance and angle calculations. Key applications include:
- Navigation and Surveying: Used in triangulation to determine the location of a point by measuring angles to it from known points.
- Astronomy: Helps in calculating the distance between planets and stars by measuring angles from different points on Earth.
- Engineering and Architecture: Used to calculate forces and angles in the design of structures like bridges, domes, and trusses to ensure stability.
