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# Law of Sines in Algebra

Using trigonometric rules, it is possible to derive various properties and values of right-angled triangles. Using the various triangle rules, it is possible to still apply trigonometry to find out various properties of non right-angled triangles such as unknown angles, lengths or area of the triangle. In this article, we will discuss one of these triangle rules - law of sines.

## The Law of Sines

The law of sines rule can be used to find missing sides or angles in a triangle.

Consider the following triangle with sides a, b and c, and angles, A, B and C.

Triangle with sides a, b and c, and angles, A, B and C, Nilabhro Datta - StudySmarter Originals

There are 2 versions of the Law of Sines.

For the above triangle, the first version of the Law of Sines states

$\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(C\right)}=\frac{c}{\mathrm{sin}\left(C\right)}$

This version of the Law of Sines is usually used to find the length of a missing side.

The second version of the Law of Sines states

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$

This version of the Law of Sines is usually used to find a missing angle.

### When to use the Law of Sines?

The law of sines can be used to solve triangles when two angles and the length of any side are known, or when the lengths of two sides and an angle opposite one of the two sides are known.

## Law of Sines Examples

For the following triangle, find a.

Solution

According to the Law of Sines,

$\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(B\right)}\phantom{\rule{0ex}{0ex}}\frac{a}{\mathrm{sin}\left({75}^{°}\right)}=\frac{8}{\mathrm{sin}\left({30}^{°}\right)}\phantom{\rule{0ex}{0ex}}\frac{a}{0.966}=\frac{8}{0.5}\phantom{\rule{0ex}{0ex}}a=15.455$

For the angle, x.

Solution

According to the Law of Sines,

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}\phantom{\rule{0ex}{0ex}}\frac{\mathrm{sin}\left(x\right)}{10}=\frac{\mathrm{sin}\left({50}^{°}\right)}{15}\phantom{\rule{0ex}{0ex}}\frac{\mathrm{sin}\left(x\right)}{10}=\frac{0.766}{15}\phantom{\rule{0ex}{0ex}}⇒x=30.71°$

## Law of sines in algebra - Key takeaways

• The law of sines rule can be used to find missing sides or angles in a triangle.
• The first version of the Law of Sines states $\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(C\right)}=\frac{c}{\mathrm{sin}\left(C\right)}$
• The second version of the Law of Sines states $\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$
• The law of sines can be used to solve triangles when two angles and the length of any side are known, or when the lengths of two sides and an angle opposite one of the two sides are known.

The Law of Sines states a/sin(A)=b/sin(B)=c/sin(C), for a triangle with sides a, b and c, and angles, A, B and C.

If two sides and an angle opposite one of the two sides is given, then the law of sines can be used to find the value of the other angle.

The law of sines can be used to solve triangles when two angles and the length of any side are known, or when the lengths of two sides and an angle opposite one of the two sides are known.

## Final Law of Sines in Algebra Quiz

Question

A triangle has two sides of length 6 and 7, and the angle opposite the side with length 7 is 72°. Find the angle opposite the side of length 6.

54.6

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Question

What is the sine rule used for?

To find missing sides or angles in a triangle.

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Question

A triangle has side lengths of 10, x and y, with angles of 30°, 85° and 65°. Use the sine rule to find the value of x and y. The side of length 10 is opposite the angle measuring 30°.

19.9 and 16.1

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