Cosine Rules

Table of Contents

Laws of Cosine

The law of cosine states that for a given triangle, the cosine of the angle opposite the longest side is equal to the sum of the cosines of the other two angles.

The law of sine states that for a given triangle, the sine of the angle opposite the longest side is equal to the ratio of the length of the opposite side to the length of the hypotenuse.

What is the Cosine Rule Formula?

The cosine rule states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The cosine rule formula is:

a^2 = b^2 + c^2 – 2bc cos A.

Laws of Cosine Trigonometric Function:

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

The cosine of an angle is also the ratio of the sine of the angle to the hypotenuse.

Cosine Rule Proof:

We’ll use the cosine rule to find the length of the side adjacent to angle A, and the length of the side opposite angle A.

We’ll start by writing out the cosine rule, with the length of the side adjacent to angle A, the length of the side opposite angle A, and angle A in parentheses:

cos(A) = adjacent / hypotenuse

We’ll solve for the length of the side adjacent to angle A:

adacent = cos(A) * hypotenuse

We’ll solve for the length of the side opposite angle A:

opposite = sin(A) * hypotenuse

We’ll substitute the values we found for adjacent and opposite into the cosine rule:

cos(A) = adjacent / hypotenuse

cos(A) = (cos(A) * hypotenuse) / hypotenuse

cos(A) = cos(A)

We’ll simplify the equation:

cos(A) = 1

Cosine Rule Proof:

To use the Cosine Rule to find the measure of angle A, we need to know the length of the side adjacent to angle A, the length of the side opposite angle A, and the length of the hypotenuse.

Let’s say that the length of the side adjacent to angle A is 6, the length of the side opposite angle A is 8, and the length of the hypotenuse is 10.

We can use the Cosine Rule to find the measure of angle A.

The Cosine Rule states that the measure of angle A is equal to the cosine of the angle A divided by the sum of the cosines of the adjacent and opposite sides.

cos(angle A) = adjacent side / hypotenuse

cos(angle A) = 6 / 10

cos(angle A) = 0.6

angle A = 30 degrees

Cosine Rule Examples:

1) Find the value of x in the equation 2×2 + 3x = 9

x = 3

2) Find the value of y in the equation y = 3×2

y = 9

Why Should You Learn the Cosine Rules?

There are a few reasons why you might want to learn the cosine rules. One reason is that they can help you to solve problems more quickly and easily. Additionally, they can help you to check your work and to ensure that you have solved a problem correctly. Finally, they can help you to better understand trigonometry and the relationships between the different trigonometric functions.

Proving the Cosine Rules

To prove the cosine rules, we will use the following definition:

The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.

To prove the cosine rule for two angles, we will use the following definition:

The cosine of two angles is the sum of the cosine of each angle.

Theorem 1:

The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.

Proof:

We will use the following diagram to help us prove the theorem.

In the diagram, we have a right triangle with angle A and angle B. We also have the length of the adjacent side, which is represented by side a, and the length of the hypotenuse, which is represented by side h.

We will start by calculating the cosine of angle A. To do this, we will divide the length of side a by the length of side h.

cos(A) = a / h

We will then do the same calculation for angle B.

cos(B) = b / h

We can now add these two values together.

cos(A) + cos(B) = a / h + b / h = 2b / h

We can now use the Pythagorean theorem to find the length of side h.

How to Study the Cosine Rules?

To study the cosine rules, first learn the basic concepts of trigonometry, including the definitions of sine, cosine, and tangent. Next, practice solving problems that use the cosine rules. There are a variety of resources available online and in print that can help you with this, including textbooks, websites, and apps. Try to find practice problems that are similar to the types of problems you will see on your test. As you become more comfortable with the cosine rules, try solving more difficult problems. Finally, review the steps involved in using the cosine rules so that you can apply them quickly and accurately when you need to.