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 AnglesCan we still apply Trigonometry to find out various properties of the given triangles, such as unknown Angles, lengths, or area? 

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Find the length of the side x.

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State the first version of the cosine rule.

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What is the cosine rule used for?

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

a² = b² + c² - 2bc·cos(A)

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What is the sine rule used for?

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

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

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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°.

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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.

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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°.

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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.

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Find the length of the side x.

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State the first version of the cosine rule.

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  • Immunology
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  • Mo

What is the cosine rule used for?

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  • Immunology
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  • Mo

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

a² = b² + c² - 2bc·cos(A)

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  • Immunology
  • Cell Biology
  • Mo

What is the sine rule used for?

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  • + Add tag
  • Immunology
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  • Mo

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

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  • Immunology
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  • Mo

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

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  • Immunology
  • Cell Biology
  • Mo

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°.

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

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.

Show Answer
  • + Add tag
  • Immunology
  • Cell Biology
  • Mo

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°.

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  • Immunology
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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.

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Contents
Table of contents

    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 sine rule - StudySmarterTriangle 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:

    asin(A)=bsin(C)=csin(C)

    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:

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

    This version of the sine rule is usually used to find a missing angle.

    For the following triangle, find a.

    Triangle Rules Finding the length of a side StudySmarter

    Solution

    According to the sine rule,

    asin(A)=bsin(B)asin(75)=8sin(30)a0.966=80.5a = 15.455

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

    For this triangle, find x.

    Triangle Rules Finding an angle StudySmarter

    Solution

    According to the sine rule,

    sin(A)a=sin(B)bsin(x)10=sin(50)15sin(x)10=0.76615x = 30.71°

    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 sine rule - StudySmarter

    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:

    cos(A) = b² + c² - a²2bc

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

    Find x.

    Triangle Rules Finding the length of a side StudySmarter

    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.

    Triangle Rules Finding an angle StudySmarter

    Solution

    According to the cosine rule,

    cos(A) = b² + c² - a²2bccos(A)=72+62-522·7·6cos(A)=57A=44.4°

    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:

    Area of a triangle = 12·base·height

    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:

    Triangle Rules area of a triangle sine rule StudySmarter

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

    Area = 12ab·sin(C) = 12bc·sin(A) = 12ac·sin(B)

    Find the area of the triangle.

    Triangle Rules area of a triangle sine rule StudySmarter

    Solution

    Area = 12ab·sin(C)= 12·6·7·sin(45)=14.85

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

    Triangle Rules Finding an angle StudySmarter

    Solution

    Area = 12ab·sin(C)10 = 12·5·8·sin(x)sin(x)=0.5x=30°

    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: asin(A)=bsin(C)=csin(C)The second version of the sine rule states thatsin(A)a=sin(B)b=sin(C)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:cos(A) = b² + c² - a²2bc
    • 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 = 12ab·sin(C) = 12bc·sin(A) = 12ac·sin(B)
    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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