Cecily Zamora
The law of sines and the law of cosines are both used to find unknown angles or sides of a triangle. The law of sines, also known as the sine rule, defines the ratio of sides of a triangle and their respective sine angles. It is used when you have two sides and the non-included angle or two angles and the non-included side. The law of cosines, on the other hand, is used when you know three sides and want to find an angle or when you know two sides, their included angle, and want to find the side length opposite the angle.
| Characteristics | Values |
|---|---|
| Law of Sines | Used to find the unknown angle or side of a triangle when two angles and one side or two angles and one included side are given |
| Used when you have two sides and the non-included angle or two angles and the non-included side | |
| Used to find unknown lengths or angles of the triangle | |
| Law of Cosines | Used to find an angle when three sides are known |
| Used when you have two sides and their included angle and you want to find the side length opposite the angle | |
| Used when you have SAS (side-angle-side) and want the third side or SSS (side-side-side) and need an angle |
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What You'll Learn
Using the Law of Sines to find unknown sides
The Law of Sines, also known as the Sine Rule, is used to find unknown sides or angles of a triangle. It defines the ratio of sides of a triangle to their respective sine angles.
The formula for the Law of Sines is:
A/sin A) = (b/sin B) = (c/sin C)
Where a, b, and c are the sides of a triangle, and A, B, and C are the angles. This formula can also be written as:
Sin A/a = b/Sin B = c/Sin C
To use the Law of Sines to find an unknown side, you need to know at least two angles and their respective side measurements. For example, if you know angles A and B, and their respective sides a and b, you can use the formula to calculate the unknown side c:
C = (b * sin C)/Sin B
The Law of Sines can also be used when you have two sides and the angle opposite them. In this case, there could be three possibilities, and you would need to check which solution makes sense in the given context.
It's important to note that the Law of Sines only applies to oblique triangles, which are triangles that are not right triangles. For right triangles, trigonometric functions such as sine, cosine, and tangent are typically used to find unknown sides or angles.
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Using the Law of Cosines to find unknown angles
The Law of Sines and the Law of Cosines are both used to solve for unknown sides and angles in a triangle. However, they are applied differently based on the given information.
The Law of Cosines can be used to find the measure of an angle when the three sides of a triangle are given. The formula for the Law of Cosines is:
> c^2 = a^2 + b^2 - 2ab cos(C)
Where 'a', 'b', and 'c' are the sides of the triangle, and 'C' is the angle between sides 'a' and 'b'. This formula is an extension of the Pythagorean Theorem that works for all triangles, not just right triangles.
To use the Law of Cosines to find an unknown angle, follow these steps:
- Identify side 'c' as the side across from the angle you are trying to find.
- Substitute the known values into the Law of Cosines formula.
- Solve the equation for angle 'C'.
For example, let's say we have a triangle with sides 'a' = 8, 'b' = 11, and 'c' = 6, and we want to find the measure of angle 'C'. We can use the Law of Cosines as follows:
> c^2 = a^2 + b^2 - 2ab cos(C)
6^2 = 8^2 + 11^2 - 2 * 8 * 11 * cos(C)
36 = 64 + 121 - 176 cos(C)
36 = 185 - 176 cos(C)
19 cos(C) = 149
Cos(C) = 149/19
Using a calculator, we can find that cos(C) = 0.798, which gives us angle C = 37 degrees.
The Law of Cosines is particularly useful when you have two sides and the included angle (SAS) and want to find the third side, or if you have all three sides (SSS) and need to find an angle. On the other hand, the Law of Sines is useful when you have a side and the opposite angle, along with another side, or when you have two angles and one side and can use basic angle subtraction to find the third angle.
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When to use the Law of Sines
The Law of Sines, or Sine Law, is used to relate two sides of a triangle to their two opposite angles. Algebraically, this means that for any pair of letters (a,b), (b,c), or (a,c), you need three of any given quadruple. For example, if you know the length of side a and the measure of angle b's opposite side, you can use the Sine Law to find the measure of angle B.
The Sine Law can also be used when you have two sides and one angle, but none of the sides are opposite the given angle. In this case, you can use basic trigonometry to find the third angle (since the angles of a triangle always add up to 180 degrees), and then use the Sine Law to solve for the remaining side.
When solving a problem, it's important to identify which angles and side lengths are given and which are being asked for. This will help determine whether to use the Sine Law or the Cosine Law.
The Sine Law is particularly useful when dealing with obtuse angles, as the function is cyclic and can provide more accurate results in these cases.
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When to use the Law of Cosines
The Law of Cosines and the Law of Sines are both used to find unknown angles or sides of a triangle. The Law of Cosines is used when three sides are known and one needs to find an angle, or when two sides and their included angle are known and one needs to find the side length opposite the angle.
The Law of Cosines formula is:
$$a^2 = b^2 + c^2 - 2bc \cdot cos( \angle a)$$
Where:
- $a$, $b$, and $c$ are the lengths of the sides of a triangle
- $\angle a$ is the angle between sides $b$ and $c"::
The Law of Cosines can be used in SAS (side-angle-side) or SSS (side-side-side) situations. For example, if you have SAS and want to find the third side, or if you have SSS and need to find an angle.
On the other hand, the Law of Sines is used when two sides and one non-included angle are known, or when two angles and one side are known, and you want to find the unknown side or angle. The Law of Sines is also known as the Sine Rule or the Sine Formula, and it defines the ratio of the sides of a triangle to their respective sine angles.
The Law of Sines formula is:
$$a/sin A = b/sin B = c/sin C$$
Where:
- $a$, $b$, and $c$ are the sides of a triangle
- $A$, $B$, and $C$ are the angles opposite their respective sides
The Law of Sines can be used in ASA (angle-side-angle) or AAS (angle-angle-side) situations.
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The Law of Sines and Cosines in trigonometry
The Law of Sines and the Law of Cosines are two trigonometric equations commonly used to find lengths and angles in scalene triangles. The Law of Sines, also known as the sine formula or sine rule, relates the lengths of a triangle's sides to the sines of its angles. The law can be expressed as:
{\displaystyle {\frac {a}{\sin {\alpha }}}\,=\,{\frac {b}{\sin {\beta }}}\,=\,{\frac {c}{\sin {\gamma }}}\,=\,2R}, 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.
The Law of Sines can be used to compute the remaining sides of a triangle when two angles and one side are known, or when two sides and the angle opposite one of them are given. In the latter case, the triangle can be divided into right triangles, which can then be solved. This technique is known as triangulation.
The Law of Cosines, on the other hand, relates the lengths of a triangle's sides to the cosine of one of its angles. The formula for the Law of Cosines is:
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma}.
The Law of Cosines can be used to find the third side of a triangle when two sides and the included angle are known, or when two sides and an angle opposite one of them are given. This theorem generalizes the Pythagorean theorem, which only applies to right triangles.
When deciding whether to use the Law of Sines or the Law of Cosines, it is important to consider the information given in the problem. The Law of Sines is useful when you have two angles and one side, or two sides and the angle opposite one of them. On the other hand, the Law of Cosines is applied when you have all three sides or two sides and their included angle.
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Frequently asked questions
The Law of Sines and Cosines are used to find the unknown angle or an unknown side of a triangle.
The Law of Sines defines the ratio of sides of a triangle and their respective sine angles, which are equivalent to each other. It is also known as the sine rule.
The Law of Cosines is used when you know three sides and want to find an angle.
