Contents

# Cosine|Definition & Meaning

**Definition**

The cosine is the **trigonometric** **ratio** of two **adjacent sides, **(the **base **and the **hypotenuse**)Â in a triangle that is calculated as the **cosine function (or cos function)**. Out of the **three** most **essential trigonometry functions**, **cosine is one** of them.

**Visual UnderstandingÂ **

Consider a right triangle **ABC**. Say side **AB** is the **base** of the triangle, Side **BC** is **perpendicular**, and side **AC** is **Hypotenuse,** then we can define Cosine as the **ratio** of Side **AB** and side **AC**. In other words, the ratio of a triangle’s **base** to its **hypotenuse**Â is referred to as **Cosine**, which is visually represented as follows.

**AB**=Base of Triangle

**BC**=Prependicular

**AC**=Hypotenuse

**$\cos\alpha=\dfrac{Base}{Hypotenuse} $**

**Mathematical Formula**

From the above figure, we know that the **Base** is **equal** to side **AB** and **Hypotenuse** is **equal** to side **AC** so we can write

**$\cos\alpha=\dfrac{AB}{AC} $**

**Properties of Cosine**

**Property 1**

In the **quadrants** the **positivity and negativity** of cosine **changes** as it moves from first to second and second to third and from third to fourth. The **positive range** for the Cosine function **lies** in **quadrant 1 and quadrant 4** whereas the **negative range** lies in **quadrants 2 and 3** as shown in the figure.

In order to verify the property, we will use a calculator.

**In Quadrant 1**

$\theta $ **ranges** from **0 to 90 degrees** so we will apply cosine to the lower limit and upper limit of quadrant 1.

**Lower Limit: **$\theta=0 $, $\cos(0)=1$

**Upper Limit: **$\theta=90$, $\cos(90)=0$

**In Quadrant 2**

$\theta $ **ranges** from greater than** 90 degrees to 180** degrees so we will apply cosine to the lower limit and upper limit of quadrant 2.

**Lower Limit: **$\theta=91 $,Â $\cos(91) = -0.0174$

**Upper Limit: **$\theta=180 $,Â $\cos(180) = -1$

**In Quadrant 3**

$\theta $ **ranges** from greater than **180 degrees** and smaller than **270 degrees** so we will apply cosine to the lower limit and upper limit of quadrant 2.

**Lower Limit: **$\theta=91 $,Â $\cos(91) = -0.0175$

**Upper Limit:Â **$\theta=269 $, $\cos(269) = -0.0174$

**In Quadrant 4**

$\theta $ **ranges** from **270 degrees to 360** degrees so we will apply cosine to the lower limit and upper limit of quadrant 2.

**Lower Limit:Â **$\theta=270 $,Â $\cos(270)=0$

**Upper Limit:Â **$\theta=360 $,Â $\cos(360)=1$

This property is visually illustrated in the figure below.

**Property 2**

The **derivative** of cosine is **equal** to the **negative** of the **sine function**. In other words, Cos is the complement of sin

$\dfrac{d}{dx}(\cos(x))=-\sin(x) $

**Property 3**

The **integration** of the **cosine **function** gives** the **sine** function.

$\int \cos(x) dx=\sin(x) + C$

**Property 4**

The cos function is a **periodic function.**

$\cos\alpha=\cos(\alpha + 2p)$

**Property 5**

The cos function is an **even function**.

$\cos(-\alpha)=\cos(\alpha)$

**Property 6**

**Trignometry formulas** for two angles, a and b, can be written as

$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$

$\cos(a-b)=\cos(a)\sin(b)+\sin(a)\cos(b)$

**Graphical Representation of Cosine Graph**

The **cos graph** starts from **90 degrees** and **repeats** itself after **every 360 degrees**.**+1 and -1** are the **maximum and minimum** values in the graph of Cos. There is a slight difference between the sin and cos graphs where the **sin graph** crosses or **sweeps** the **origin** whereas the **cos graph** **never** does so. The graph for cosine is shown in the figure below.

**Properties of Cos Graph**

- The graph of
**cosine never sweeps**from the origin point. - For every positive and negative value of theta, the
**cos graph****maintains**its**continuity**and**repeats**after e**very 360 degrees**. - The
**amplitude**of the**cos graph**can be referred to as**1**with**positive 1**being the**maximum value**and**negative 1**being the**smallest value**.

**Law of Cosines**

A** triangle’s side** lengths are **related to** one of its **angles** **through** the **law of cosines**. The use of **trigonometry** allows us to** measure distances** and **angles** that can’t be calculated any other way. When we **know two sides** and the enclosed **angle** of the triangle, we can **apply **the law of **cosines** to **compute** the **third side**.

In a triangle, the **cosine** of its **angles** and the **length** of its sides can be **determined by** the **law of cosine**. In trigonometry, a **right triangle** is **represented** by the **cosine law**, which generalizes **Pythagoras’ theorem**.

**Statement**

**Consider** a **triangle** with sides **x, y, and z. **Then the **square of one of the sides,** say z, is **equal** to the **difference between the sum of squares of side** x, and side y and **two times the product of side** x and side y and angle between them.

**$z^{2}=x^{2}+y^{2}-2xy\cos(Z)$**

The **cosine rule** is **another name** for the **law of cosine.** **Any triangle** can be **solved** with the **help** of **this law**. This rule can be used, for example, to **find** the **third side** of a triangle when you know the **lengths** **of two sides** and the **degree** included **between them**. The illustration is shown in the figure.

**Formula**

There are **three laws of cosines**. Consider a triangle having three sides x, y and z. In order to find:

**Side x, when side y, z is given along angle X**

\[x^{2}=y^{2}+z^{2}-2yz\cos(X)\]

**Side y, when side x, z is given along angle Y**

\[y^{2}=x^{2}+z^{2}-2xz\cos(Y)\]

**Side z, when side y, z is given along angle Z**

\[z^{2}=x^{2}+y^{2}-2xy\cos(X)\]

**Properties**

- The Law of cosines is used to
**find an unknown side**of triangle when**two other**triangles are**given**. - The Law of cosines is used to
**find the angle**between the sides when all**three sides**are**given**. - The Law of cosines is not limited to just right-angle triangles it can be
**easily applied**to**any other triangle**whose sides or angles are to be found.

**Solved Examples With the Cosine Function**

**Example 1**

**Consider** **four values** of **theta** and **compute** the **cos function** for all the values of theta using the calculator.

**Solution**

$\theta=0,90,180,270,360$

$\cos(0)=1$

$\cos(90)=0$

$\cos(270)=0$

$\cos(360)=1$

**Example 2**

Consider a triangle ABC with side **AC=2 AB=5 BC=$\sqrt2$** **Find** the angle **$\alpha$**. The figure from this problem is shown below.

**Solution**

$\cos\alpha=\dfrac{Base}{Hypotenuse} $

$\cos\alpha=\dfrac{BC}{AB} $

$\alpha=\arccos\dfrac{\sqrt2}{5} $

**$\alpha=73.57^{\circ}$**

**Example 3**

Consider the following triangle with side **AB=4 BC=5 AC=6** and **finds the angle A**. The figure from this problem is shown below.

**Solution**

Applying the law of cosines:

$a^{2}=b^{2}+c^{2}-2bc\cos(A)$

$5^{2}=6^{2}+4^{2}-2.6.4\cos(A)$

$25=36+16-48 x \cos(A)$

$-48\cos(A)=-27$

$\cos(A)=\dfrac{-27}{-48}$

$\cos(A)=0.5625$

$A=\arccos0.5625$

**$A=55.77^{\circ}$**

*All mathematical drawings and images were created with GeoGebra.*