The triangle rules discussed in this article will explore this question in further detail:

## Triangle rules – sine rule

The first triangle rule that we will discuss is called the sine rule. The sine 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.

There are two versions of the sine rule.

For the above triangle, the first version of the sine rule 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 sine rule is usually used to find the length of a missing side.**

The second version of the sine rule 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 sine rule is usually used to find a missing angle.**

For the following triangle, find a.

#### Solution

According to the sine rule,

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

Read Sine and Cosine Rules to learn about the sine rule in greater depth.

For this triangle, find x.

#### Solution

According to the sine rule,

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}\phantom{\rule{0ex}{0ex}}\frac{\mathrm{sin}\left(x\right)}{10}=\frac{\mathrm{sin}\left(50\right)}{15}\phantom{\rule{0ex}{0ex}}\frac{\mathrm{sin}\left(x\right)}{10}=\frac{0.766}{15}\phantom{\rule{0ex}{0ex}}\Rightarrow x=30.71\xb0$

## Triangle rules – cosine rule

The second triangle rule that we will discuss is called the cosine rule. The cosine 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 two versions of the cosine rule.

For the above triangle, the first version of the cosine rule states:

a² = b² + c² - 2bc · cos (A)

**This version of the cosine rule is usually used to find the length of a missing side when you know the lengths of the other two sides and the angle between them.**

The second version of the cosine rule states:

$\mathrm{cos}\left(A\right)=\frac{b\xb2+c\xb2-a\xb2}{2bc}$

**This version of the cosine rule is usually used to find an angle when the lengths of all three sides are known.**

Find x.

#### Solution

According to the cosine rule,

a² = b² + c² - 2bc · cos (A)

=> x² = 5² + 8² - 2 x 5 x 8 x cos (30)

=> x² = 19.72

=> x = 4.44

For the next triangle, find angle A.

#### Solution

According to the cosine rule,

$\mathrm{cos}\left(A\right)=\frac{b\xb2+c\xb2-a\xb2}{2bc}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{cos}\left(A\right)=\frac{{7}^{2}+{6}^{2}-{5}^{2}}{2\xb77\xb76}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{cos}\left(A\right)=\frac{5}{7}\phantom{\rule{0ex}{0ex}}\Rightarrow A=44.4\xb0$

Read Sine and Cosine Rules to learn about the cosine rule in greater depth.

## Triangle Rules – the area of a triangle

We are already familiar with the following formula:

$Areaofatriangle=\frac{1}{2}\xb7base\xb7height$

But what if we do not know the exact height of the triangle? We can also find out the area of a triangle for which we know the **length of any two sides and the angle between them**.** **

Consider the following triangle:

The area of the above triangle can be found by using the formula:

$Area=\frac{1}{2}ab\xb7\mathrm{sin}\left(C\right)=\frac{1}{2}bc\xb7\mathrm{sin}\left(A\right)=\frac{1}{2}ac\xb7\mathrm{sin}\left(B\right)$

Find the area of the triangle.

#### Solution

$Area=\frac{1}{2}ab\xb7\mathrm{sin}\left(C\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\xb76\xb77\xb7\mathrm{sin}\left(45\right)\phantom{\rule{0ex}{0ex}}=14.85$

The area of the triangle is 10 Units. Find the angle x.

#### Solution

$Area=\frac{1}{2}ab\xb7\mathrm{sin}\left(C\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 10=\frac{1}{2}\xb75\xb78\xb7\mathrm{sin}\left(x\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{sin}\left(x\right)=0.5\phantom{\rule{0ex}{0ex}}\Rightarrow x=30\xb0$

Click on Area of Triangles to learn about the area of triangles rule in greater depth.

## Triangle rules – key takeaways

- You can use the sine rule to find missing sides or angles in a triangle.
- The first version of the sine rule states that: $\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(C\right)}=\frac{c}{\mathrm{sin}\left(C\right)}$The second version of the sine rule states that$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$
- You can use the cosine rule to find missing sides or angles in a triangle.
- The first version of the cosine rule states that:a² = b² + c² - 2bc · cos (A) The second version of the sine rule states that:$\mathrm{cos}\left(A\right)=\frac{b\xb2+c\xb2-a\xb2}{2bc}$
- We can find out the area of a triangle for which we know the length of any two sides and the angle between them using the following formula:$Area=\frac{1}{2}ab\xb7\mathrm{sin}\left(C\right)=\frac{1}{2}bc\xb7\mathrm{sin}\left(A\right)=\frac{1}{2}ac\xb7\mathrm{sin}\left(B\right)$

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##### Frequently Asked Questions about Triangle Rules

What is the sine rule for triangles?

The sine rule for triangles states that

a/sin(A)=b/sin(B)=c/sin(C)

How do you calculate area of a triangle using the sine rule?

The sine rule can be used to find find missing sides or angles in a triangle. Once we have sufficient information, we can used the formula Area=1/2*a*b*sin(C)

Can the sine rule be used with right-angled triangles?

Yes, in that case, one of the angles will be 90.

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