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## The Law of Cosines

For any triangle:

## How to Remember

How can you remember the formula?

Well, it helps to know it’s the Pythagoras Theorem with something extra so it works for all triangles:

Pythagoras Theorem:

(only for Right-Angled Triangles)a^{2} + b^{2} = c^{2}Law of Cosines:

(for all triangles)a^{2} + b^{2} − 2ab cos(C) = c^{2}

So, to remember it:

- think “
**abc**“:**a**^{2}+**b**^{2}=**c**^{2}, - then a
**2**nd “**abc**“:**2ab**cos(**C**), - and put them together:
**a**^{2}+ b^{2}− 2ab cos(C) = c^{2}

## When to Use

The Law of Cosines is useful for finding:

- the third side of a triangle when we know
**two sides and the angle between**them (like the example above) - the angles of a triangle when we know
**all three sides**(as in the following example)

## In Other Forms

### Easier Version For Angles

We just saw how to find an angle when we know three sides. It took quite a few steps, so it is easier to use the “direct” formula (which is just a rearrangement of the c^{2} = a^{2} + b^{2} − 2ab cos(C) formula). It can be in either of these forms:

### Versions for a, b and c

Also, we can rewrite the c^{2} = a^{2} + b^{2} − 2ab cos(C) formula into a^{2}= and b^{2}= form.

Here are all three:

a^{2} = b^{2} + c^{2} − 2bc cos(A)

b^{2} = a^{2} + c^{2} − 2ac cos(B)

c^{2} = a^{2} + b^{2} − 2ab cos(C)

But it is easier to remember the “**c ^{2}**=” form and change the letters as needed !

As in this example:

## Cosine Rule

In trigonometry, the **Cosine Rule** says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them. **Cosine rule is also called law of cosines** or **Cosine Formula.**

Suppose, a, b and c are lengths of the side of a triangle ABC, then;

where ∠x, ∠y and ∠z are the angles between the sides of the triangle.

The cosine rule relates to the lengths of the sides of a triangle with any of its angles being a cosine angle. With the help of this rule, we can calculate the length of the side of a triangle or can find the measure of the angle between the sides.

## What are the Laws of Cosine?

As per the diagram, Cosine rules to find the length of the sides a, b & c of the triangle ABC is given by;

- a
^{2}= b^{2}+ c^{2}– 2bc cos x - b
^{2}= a^{2}+ c^{2}– 2ac cos y - c
^{2}= a^{2}+ b^{2}– 2ab cos z

Similarly, to find the angles x, y and z, these formulae can be re-written as :

- cos x = (b
^{2}+ c^{2}-a^{2})/2bc - cos y = (a
^{2}+ c^{2}-b^{2})/2ac - cos z = (a
^{2}+ b^{2}– c^{2})/2ab

## Proof

The law of cosine states that for any given triangle say ABC, with sides a, b and c, we have;

c^{2} = a^{2} + b^{2} – 2ab cos C

Now let us prove this law.

Suppose a triangle ABC is given to us here. From the vertex of angle B, we draw a perpendicular touching the side AC at point D. This is the height of the triangle denoted by h.

Now in triangle BCD, as per the trigonometry ratio, we know;

cos C = CD/a [cos θ = Base/Hypotenuse]

or we can write;

CD = a cos C ………… (1)

Subtracting equation 1 from side b on both the sides, we get;

b – CD = b – a cos C

or

DA = b – a cos C

Again, in triangle BCD, as per the trigonometry ratio, we know;

sin C = BD/a [sin θ = Perpendicular/Hypotenuse]

or we can write;

BD = a sin C ……….(2)

Now using Pythagoras theorem in triangle ADB, we get;

c^{2} = BD^{2} + DA^{2} [Hypotenuse^{2} = Perpendicular^{2} + Base^{2} ]

Substituting the value of DA and BD from equation 1 and 2, we get;

c^{2} = (a sin C)^{2} + (b – a cos C)^{2}

c^{2} = a^{2} sin^{2}C + b^{2} – 2ab cos C + a^{2} cos^{2} C

c^{2} = a^{2} (sin^{2}C + cos^{2} C) + b^{2} – 2ab cos C

By trigonometric identities, we know;

sin^{2}θ+ cos^{2}θ = 1

Therefore,

c^{2} = a^{2} + b^{2} – 2ab cos C

Hence, proved.

## Cosine Formula

The formula to find the sides of the triangle using cosine rule is given below:

### Sine Formula

As per sine law,

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 respective angles.

Also, we can write:

a: b: c = Sin A: Sin B: Sin C

## Solved Example

Find the length of x in the following figure.

**Solution: **By applying the **Cosine rule**, we get:

x^{2} = 22^{2} +28^{2} – 2 x 22 x 28 cos 97

x^{2} = 1418.143

x = √ 1418.143

Learn more about different Math topics with BYJU’S – The Learning App

## Frequently Asked Questions – FAQs

### What is the cosine rule for a triangle?

According to the Cosine Rule, the square of the length of any one side of a triangle is equal to the sum of the squares of the length of the other two sides subtracted by twice their product multiplied by the cosine of their included angle. Mathematically it is given as:

a^{2} = b^{2} + c^{2} – 2bc cos x

### When can we use the cosine rule?

We can use the cosine rule,

Either, when all three sides of the triangle are given and we need to find all the angles

Or, to find the third side of the triangle when two sides and the angle between them are known.

### How to find angles using the cosine rule?

If C is the included angle, then

Cos C = (a^{2} + b^{2} – c^{2})/2ab

Where a,b and c are the sides of the triangle.

### What is the cosine formula?

The cosine formula to find the side of the triangle is given by:

c = √[a^{2} + b^{2} – 2ab cos C]

Where a,b and c are the sides of the triangle.

### What is the sine rule formula?

As per the sine rule, if a, b, and c are the length of sides of a triangle and A, B, and C are the angles, then,

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

## Cosine Formulas

The cosine formulas are formulas of the cosine function in trigonometry. The cosine function (which is usually referred to as “cos”) is one of the 6 trigonometric functions which is the ratio of the adjacent side to the hypotenuse. There are multiple formulas related to cosine function which can be derived from various trigonometric identities and formulas. Let us learn the cosine formulas along with a few solved examples.

## What Are Cosine Formulas?

The cosine formulas talk about the cosine (cos) function. Let us consider a right-angled triangle with one of its acute angles to be x. Then the cosine formula is, cos x = (adjacent side) / (hypotenuse), where “adjacent side” is the side adjacent to the angle x, and “hypotenuse” is the longest side (the side opposite to the right angle) of the triangle. Apart from this general formula, there are so many other formulas in trigonometry that will define cosine function which you can see in the following image.

### Cosine Formulas Using Reciprocal Identity

We know that the cosine function (cos) and the secant function (sec) are reciprocals of each other. i.e., if cos x = a / b, then sec x = b / a. Thus, cosine formula using one of the reciprocal identities is,

cos x = 1 / (sec x)

### Cosine Formulas Using Pythagorean Identity

One of the trigonometric identities talks about the relationship between sin and cos. It says, sin^{2}x + cos^{2}x = 1, for any x. We can solve this for cos x.

Consider sin^{2}x + cos^{2}x = 1

Subtracting sin^{2}x from both sides,

cos^{2}x = 1 – sin^{2}x

Taking square root on both sides,

cos x = ± √(1 – sin^{2}x)

### Cosine Formula Using Cofunction Identities

The cofunction identities define the relation between the cofunctions which are sin, cos; sec, csc, tan, and cot. Using one of the cofunction identities,

- cos x = sin (90
^{o}– x) (OR) - cos x = sin (π/2 – x)

### Cosine Formulas Using Sum/Difference Formulas

We have sum/difference formulas for every trigonometric function that deal with the sum of angles (x + y) and the difference of angles (x – y). The sum/difference formulas of cosine function are,

- cos(x + y) = cos (x) cos(y) – sin (x) sin (y)
- cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

### Cosine Formula of Double Angle

We have double angle formulas in trigonometry which deal with 2 times the angle. We have multiple double angle formulas of cos and we can use one of the following while solving the problem depending on the available information. They are,

- cos 2x = cos
^{2}(x) – sin^{2}(x) - cos 2x = 2 cos
^{2}(x) − 1 - cos 2x = 1 – 2 sin
^{2}(x) - cos 2x = [(1 – tan
^{2}x)/(1 + tan^{2}x)]

### Cosine Formula of Triple Angle

We have triple angle formulas for all trigonometric functions. Among them, the triple angle formula of the cosine function is,

cos 3x = 4 cos^{3}x – 3 cos x

### Cosine Formula of Half Angle

We have half-angle formulas in trigonometry that deal with half of the angles (x/2). The half-angle formula of the cosine function is,

cos (x/2) =± √[ (1 + cos x) / 2 ]

### Cosine Formulas Using Law of Cosines

The law of cosines is used to find the missing sides/angles in a non-right angled triangle. Consider a triangle ABC in which AB = c, BC = a, and CA = b. The cosine formulas using the law of cosines are,

- cos A = (b
^{2}+ c^{2}– a^{2}) / (2bc) - cos B = (c
^{2}+ a^{2}– b^{2}) / (2ac) - cos C = (a
^{2}+ b^{2}– c^{2}) / (2ab)

## Examples on Cosine Formulas

**Example 1: **If sin x = 3/5 and x is in the first quadrant, find the value of cos x.

**Solution:**

Using one of the cosine formulas,

cos x = ± √(1 – sin^{2}x)

Since x is in the first quadrant, cos x is positive. Thus,

cos x = √(1 – sin^{2}x)

Substitute sin x = 3/5 here,

cos x = √(1 – (3/5)^{2})

= √(1 – 9/25)

=√ (16/25)

= 4/5

**Answer: **cos x = 4/5.

**Example 2: **If sin (90 – A) = 1/2, then find the value of cos A.

**Solution:**

Using one of the cosine formulas,

cos A = sin (90 – A)

It is given that sin (90 – A) = 1/2. Hence,

cos A = 1/2

**Answer: **cos A = 1/2.

**Example 3: **In a triangle ABC, AB = c, BC = a, and CA = b. Also, a = 55 units, b = 70 units, and c = 50 units. Find cos A.

**Solution:**

Using the cosine formula of law of cosines,

cos A = (b^{2} + c^{2} – a^{2}) / (2bc)

= (70^{2} + 50^{2} – 55^{2}) / (2 · 70 · 50)

= 5/8

**Answer: **cos A = 5/8.

FAQs on Cosine Formulas

### What Are Cosine Formulas?

The cosine formulas are related to the cosine function. The important cosine formulas are as follows:

- cos x = (adjacent side) / (hypotenuse)
- cos x = 1 / (sec x)
- cos x = ± √(1 – sin
^{2}x) - cos x = sin (π/2 – x)

### How To Derive Half Angle Cosine Formula?

Using one the double angle formulas, cos 2x = 2 cos^{2} x – 1. Replacing x with (x/2) on both sides, we get cos x = 2 cos^{2} (x/2) – 1. Solving this for cos (x/2), we get cos (x/2) =± √[ (1 + cos x) / 2 ].

### What Are the Applications of Cosine Formulas?

As we have learned on this page, we have multiple cosine formulas and we can choose one of them to prove a trigonometric identity (or) find the value of the cosine function with the available information. We can use the cosine formulas to find the missing angles or sides in a triangle. We also use cosine formulas in Calculus.

### How to Derive the Double Angle Cosine Formula?

Using the sum formula of cosine function, we have, cos(x + y) = cos (x) cos(y) – sin (x) sin (y). Substituting x = y on both sides here, we get, cos 2x = cos^{2}x – sin^{2}x. Using the Pythagorean identity sin^{2}x + cos^{2}x = 1, along with the above formula, we can derive two other double angle cosine formulas which are cos 2x = 2 cos^{2}(x) − 1 and cos 2x = 1 – 2 sin^{2}(x).

## Cosine rule examples (missing side)

Find the value of x for triangle ABC, correct to 2 decimal places.

z= 0.504mm (3dp)

θ=76^{∘} (2sf)

θ=102.04^{∘} (2dp)

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

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