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# Triangle Rules

It is possible to derive various properties and values of right-angled triangles using trigonometric rules. But what if we are dealing with triangles that don't have any right angles? Can we still apply trigonometry to find out various properties of the given triangles, such as unknown angles, lengths, or area?

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.

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

There are two versions of the sine rule.

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

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:

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,

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,

## 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:

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,

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:

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:

Find the area of the triangle.

#### Solution

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

#### Solution

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: The second version of the sine rule states that
• 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:
• 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:

## Frequently Asked Questions about Triangle Rules

The sine rule for triangles states that

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

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)

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

## Final Triangle Rules Quiz

Question

What is the sine rule used for?

The sine rule is used to find missing sides or angles in a triangle.

Show question

Question

Find the length of the side x.

15.32

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Question

State the first version of the cosine rule.

a² = b² + c² - 2bc·cos(A)

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Question

What is the cosine rule used for?

The cosine rule can be used to find missing sides or angles in a triangle.

Show question

Question

When is the following version of the cosine rule usually used?

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.

Show question

Question

State whether the following statement is true or false: The two versions of the sine rule are equivalent to each other.

True

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Question

State whether the following statement is true or false: The two versions of the cosine rule are equivalent to each other.

True

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Question

We have a triangle of side lengths 12, 13 and 4. Use the cosine rule to find the angle sizes in the triangle.

[Hint: Use the cosine rule three times]

66.8°, 17.8° and 95.4°

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Question

A triangle has side lengths 10, 4 and x. The angle opposite x is 100°. Find the value of x.

11.4

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Question

A triangle has side lengths 2, 8 and z. The angle opposite x is 18°. Find the value of x.

6.13

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Question

True or false: The angle used in the cosine rule must be acute.

False

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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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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

A triangle has two angles measuring 67° and 33° respectively. The side opposite the angle measuring 67° has a length of 9. Find the length of the side opposite the angle measuring 33°.

5.33

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Question

A triangle has two sides of length 13 and 7, and the angle opposite the side with length 13 is 144°. Find the angle opposite the side of length 7.

18.5°

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Question

A triangle has two angles measuring 58° and 23° respectively. The side opposite the angle measuring 58° has a length of 15. Find the length of the side opposite the angle measuring 23°.

6.91

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Question

What units would you use to measure the area of this shape?

unit2

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