Cecily Zamora
The Law of Sines and the Law of Cosines are trigonometric formulas used to solve for unknown sides or angles in a triangle. The sine rule lets you relate two sides and their two opposite angles, whereas the cosine rule is used when you know the lengths of two sides and the measurement of the angle between them. The Law of Cosines can be useful when you have two sides and the angle between them (SAS) and want to find the third side, or if you have the lengths of all three sides (SSS) and need to find an angle. The Law of Sines can be used when you have a side and its opposite angle and another side, or when you have two angles and one side.
| Characteristics | Values |
|---|---|
| Relating Sides and Angles | The sine law relates two sides and their two opposite angles. The cosine law relates three sides and any one angle. |
| Given Information | The sine law is used when you have a side, its opposite angle, and another side. The cosine law is used when you have two sides and one angle that is not opposite to either side. |
| Finding Unknowns | The sine law is used to find the unknown angle. The cosine law is used to find the third side when SAS (side-angle-side) is known or to find an angle when SSS (side-side-side) is known. |
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What You'll Learn
When to use the sine law
The sine rule or sine law can be used in any triangle, not just right-angled triangles, where a side and its opposite angle are known. The sine law lets you relate two sides and their two opposite angles.
When solving a problem, look at which angles and side lengths are given, and which are being asked for. If you have a side and its opposite angle, and another side, you can use the sine rule and then find the angle with sin inverse. If you have two sides and one angle, but none of the sides are opposite the given angle, you must use the cosine rule.
The sine rule is useful when you have two sides and an angle that isn't opposite to either side. For example, if you have sides a and b, and angle C, you can use the sine rule to find angle A.
The sine rule formula will contain two fractions, each with a side and its opposite angle. You will only ever need two parts of the formula, not all three. It is important to keep full accuracy until the end of your calculation to avoid errors.
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When to use the cosine law
The law of cosines, also known as the cosine formula, cosine rule, or théorème d'Al-Kashi, is a trigonometric rule that relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used to solve mathematical problems, particularly when trying to find unknown values in an oblique triangle, or a triangle without a right angle.
The law of cosines can be used when you know the values of two sides and the included angle (SAS) or when you know the values of three sides and want to work out an angle (SSS).
A^2 = b^2 + c^2 - 2bccos(A)
The formula can be rearranged to solve for different known values. For example, to find a missing side, you can use:
Cos(A) = (b^2 + c^2 - a^2) / 2bc
The law of cosines is a more general formula than the law of sines, as it works for all types of triangles, not just right triangles. It can be used to find the third side of a triangle if two sides and the angle between them are known, or if two sides and an angle opposite to one of them are known.
In summary, the law of cosines is useful when you have SAS and want the third side, or if you have SSS and need an angle.
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Using the sine law with right triangles
The sine rule, or the law of sines, is a trigonometric mathematical equation that can be used to compute the sides of a triangle when two angles and one side are known. The law of sines can be used for any triangle, not just right-angled triangles, as long as a side and its opposite angle are known.
The law of sines states:
>
Where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles, while R is the radius of the triangle's circumcircle.
When the last part of the equation is not used, the law can be stated using reciprocals:
>
To use the sine rule with right triangles, you can follow these steps:
- Identify the given information: Start by identifying the values of the two angles and one side that are given in the right triangle. Let's denote the given side as 'a' and the given angles as 'α' and 'β'.
- Apply the sine rule formula: Use the sine rule formula to set up an equation relating the given information to the unknown side or angle. Depending on the specific values given, you can manipulate the formula accordingly.
- Solve for the unknown: Use trigonometric functions or algebraic manipulations to solve for the unknown side or angle. Ensure that you maintain full accuracy throughout the calculation to avoid errors.
- Validate the solution: Check that the solution obtained satisfies the given conditions and constraints of the problem. Ensure that the solution makes sense in the context of the right triangle.
- Interpret the results: Finally, interpret the results in the context of the problem. You may need to provide additional information or conclusions based on the calculated values.
For example, let's say we have a right triangle with a hypotenuse of 5 units and one angle of 30 degrees. We want to find the length of the side opposite the 30-degree angle.
- Given information: We have 'a' = 5 units (hypotenuse) and 'α' = 30 degrees. We want to find the side opposite 'α'.
- Apply the sine rule: Using the reciprocals form of the sine rule, we can set up the equation:
>
Where 'b' is the side we want to find.
Solve for the unknown: We can rearrange the equation to solve for 'b':
>
Now, we can plug in the values:
>
Calculating the sine of 30 degrees and assuming the angle 'α' is in the same unit, we can find the value of 'b'.
Validate and interpret: Finally, we validate that the solution satisfies the given conditions and interpret the result in the context of the problem.
The law of sines provides a useful technique for solving right triangles, especially when two angles and a side are known. By following the steps outlined above, you can effectively apply the sine rule to solve for unknown sides or angles in right triangles.
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Using the cosine law with right triangles
The law of cosines, also called the cosine rule, states that for any triangle with sides a, b, and c, and angle C opposite side c:
C^2 = a^2 + b^2 − 2ab cos(C)
This formula can be used to find the third side of a triangle when we know two sides and the angle between them, or to find the angles of a triangle when we know all three sides. For example, if we have a triangle with sides of length 8 and 11, and an angle of 37 degrees between them, we can use the law of cosines to find the length of the third side:
C^2 = 8^2 + 11^2 − 2 * 8 * 11 * cos(37°)
C^2 = 64 + 121 − 176 * 0.798...
C^2 = 44.44...
C = √44.44 = 6.67 to 2 decimal places
The law of cosines is particularly useful for solving problems involving right triangles. In a right triangle, the Pythagorean theorem states that a^2 + b^2 = c^2, where c is the length of the hypotenuse. The law of cosines is a more general form of the Pythagorean theorem that can be used for all triangles, not just right triangles.
For example, consider a right triangle with legs of length 3 and 4. The hypotenuse can be calculated using the Pythagorean theorem:
C^2 = 3^2 + 4^2
C^2 = 9 + 16
C^2 = 25
C = 5
Now consider a non-right triangle with sides of length 3, 4, and 5. The angle between the sides of length 3 and 4 is acute, so we can use the law of cosines to find the measure of the included angle:
5^2 = 3^2 + 4^2 − 2 * 3 * 4 * cos(C)
25 = 9 + 16 − 24 * cos(C)
9 = 24 * cos(C)
Cos(C) = 9/24
Cos(C) = 0.375
C = cos^(-1)(0.375)
C = 68.2°
In summary, the law of cosines is a powerful tool for solving problems involving right and non-right triangles. It can be used to find missing sides or angles when given the appropriate input information. The law of cosines is a generalisation of the Pythagorean theorem and can be used for all triangles, making it a valuable tool in trigonometry and geometry.
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How the laws differ
The sine and cosine laws are used to solve for unknown sides or angles in a triangle. They differ in the number of sides and angles required to be known in order to use them.
The sine law relates two sides of a triangle to their two opposite angles. That is, if you know a side and its opposite angle, and another side, you can use the sine law to find the angle between the two known sides.
The cosine law, on the other hand, relates three sides of a triangle to one of its angles. So, if you know the lengths of two sides and the angle between them, and want to find the length of the third side, you would use the cosine law.
The sine law is also useful when you know two angles and one side, as you can subtract to find the third angle, and then use the sine law to find the unknown sides.
In summary, the sine law is used when you know two sides and their opposite angles, or when you know two angles and one side. The cosine law is used when you know two sides and the angle between them, or when you know the lengths of all three sides and want to find an angle.
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Frequently asked questions
The Law of Sines lets you relate two sides and their two opposite angles. The Law of Cosines is used when you know the lengths of two sides and the measurement of the angle between them.
If you have a side and an opposite angle and another side, you can use the Law of Sines and then find the angle with sin inverse. If you have two sides and one angle, but none of the sides are opposite to the given angle, you have to use the Law of Cosines.
When solving a problem, look at what angles and side lengths are given and what side length/angle the problem asks you to find. Then choose a formula that uses the given angles/side lengths and the ones you are trying to find. The Law of Cosines is good when you have two sides and an angle (SAS) and want the third side, or if you have the lengths of all three sides (SSS) and need an angle. The Law of Sines is good for the other situations.
The Law of Cosines can be shown as a formula that uses the Pythagorean Theorem.
