About Law of Sines
The law of sines provides the link between an oblique triangle's sides and angles (non-right triangle). In trigonometry, the laws of sines and cosines are key rules for "solving a triangle." The sine rule states that the ratios of a triangle's side lengths to the sine of its respective opposite angles are equal.
What do you mean by Law of Sines?
The law of sines connects the ratios of triangle side lengths to their opposing angles. For all three sides and opposite angles, this ratio remains constant. Using the needed known data, we can use the sine rule to find the missing angle or side of any triangle.
Law of Sines
The diameter of the circumcircle of a triangle is equal to the ratio of the side and the corresponding angle. As a result, the sine law can be written as,
a/sinA = b/sinB = c/sinC = 2R
- The lengths of the triangle's sides are a, b, and c.
- The triangle's angles are A, B, and C.
- The radius of the triangle's circumcircle is R.
Law of Sines Formula
For linking the lengths of a triangle's sides to the sines of consecutive angles, the law of sines formula is utilised. It is the ratio of the length of one of the triangle's sides to the sine of the angle generated by the remaining two sides. Apart from SAS and SSS triangles, the law of sines formula is applied to any triangle. It says,
a/sin A = b/sin B = c/sin C
where a, b, and c are the triangle's lengths and A, B, and C are the triangle's angles. This formula can be written in three different ways, as follows:
a/sinA = b/sinB = c/sinC
sinA/a = sinB/b = sinC/c
a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC
Download free pdf of Law of Sines Its Use And Solved Examples
Frequently Asked Questions
The Law of Sines is a mathematical formula that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it’s expressed as:
asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}sinAa=sinBb=sinCc
Where aaa, bbb, and ccc are the lengths of the sides, and AAA, BBB, and CCC are the angles opposite these sides.
The Law of Sines is used when:
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You are given two angles and one side of a triangle (ASA or AAS).
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You are given two sides and a non-included angle (SSA).
This law is particularly useful in solving non-right-angled (oblique) triangles.
Yes, the Law of Sines can be used for any type of triangle—whether it is acute, obtuse, or right-angled. However, for right-angled triangles, the Law of Cosines is often more convenient. The Law of Sines is especially useful for solving non-right-angled triangles.
The Law of Sines is used in various fields such as:
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Surveying: To measure distances and angles between inaccessible points.
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Astronomy: For calculating angles and distances between celestial objects.
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Navigation: To calculate the course of ships or aircraft based on their positions and angles.
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Engineering: In structural analysis and design where triangular shapes are involved.
The Ambiguous Case occurs when you are given two sides and an angle (SSA condition) in a triangle. In this case, there could be:
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One solution (a unique triangle),
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Two possible solutions (two different triangles),
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No solution (no triangle can be formed).
This ambiguity arises because the given information might correspond to more than one triangle, or no triangle at all.
Yes, the Law of Sines applies to both acute and obtuse triangles. However, in right triangles, it is less commonly used compared to the Pythagorean Theorem or the Law of Cosines, as these are more straightforward for right-angled triangles.
Yes, the Law of Sines is applicable to triangles with obtuse angles. The formula works for all types of triangles (acute, obtuse, and right), provided you are given sufficient information (such as two angles and one side, or two sides and a non-included angle).
Both the Law of Sines and the Law of Cosines are used to solve triangles, but they are applied in different scenarios:
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The Law of Sines is often used when you know two angles and one side, or two sides and a non-included angle.
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The Law of Cosines is used when you know two sides and the included angle, or all three sides of the triangle.