Law of Sines and Cosines -

The Law of Sines and the Law of Cosines are two essential formulas in trigonometry that help solve triangles, including those that are not right-angled.

Law of Sines:
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. Mathematically, it can be expressed as follows:

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

In this formula, ‘a, ‘b’, and ‘c’ represent the lengths of the sides of the triangle, while ‘A’, ‘B’, and ‘C’ represent the opposite angles.

Here’s an example to illustrate the Law of Sines:

Consider a triangle with side lengths a = 5, b = 7, and angle C = 40 degrees. To find the length of side c, we can use the Law of Sines.

$c/sin(C) = a/sin(A)$
$c/sin(40°) = 5/sin(A)$

If we solve for ‘c’ by cross-multiplying and rearranging the equation, we get:

$c = (5 * sin(40°)) / sin(A)$

Similarly, we can use the Law of Sines to find the length of side b:

$b/sin(B) = a/sin(A)$
$7/sin(B) = 5/sin(A)$

Again, solving for ‘b’, we get:

$b = (7 * sin(A)) / sin(B)$

Law of Cosines:
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is beneficial when you have enough information about the triangle but do not know any of the angles. The formula is as follows:

$c^2 = a^2 + b^2 - 2ab * cos(C)$

In this formula, ‘a’, ‘b’, and ‘c’ represent the lengths of the sides of the triangle, while ‘C’ represents the angle between sides ‘a’ and ‘b’.

Let’s use the Law of Cosines to find an unknown side length in a triangle:

Consider a triangle with side lengths a = 4, b = 6, and angle C = 50 degrees. We can use the Law of Cosines to find the length of side c.

$c^2 = a^2 + b^2 - 2ab * cos(C)c^2 = 4^2 + 6^2 - 2(4)(6) * cos(50°)$

Simplifying the equation further:

$c^2 = 16 + 36 - 48cos(50°)$

Finally, we can find ‘c’ by taking the square root of both sides of the equation.

$c = √(16 + 36 - 48cos(50°))$

Here is a diagram to help visualize the triangle in the Law of Sines and Law of Cosines examples:


         A
        / \
   c   /     \   b
      /        \
     /________\
   B       a        C

In the diagram, ‘a’, ‘b’, and ‘c’ represent the lengths of the sides of the triangle, while ‘A’ and ‘B’ represent the angles opposite to sides ‘a’ and ‘b’, respectively.

Law of Sines and Cosines

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