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by William Smith

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The **cos** **45 degrees**, symbolized as **cos(45°)** holds a special place in the fascinating world of **trigonometry**, we encounter key concepts related to **cos(45°) **that serve as the foundation of this mathematical discipline. The **cosine** of an angle is a fundamental measure that plays a vital role in many mathematical and physical phenomena.

This discussion focuses on a specific, highly significant angle: **45 degrees**. The **cosine** of **45 degrees**, symbolized as **cos(45°)**, carries an intriguing property of **equanimity** due to its equidistant position on the **unit circle** and is deeply embedded in various mathematical applications.

From **geometry** to **physics**, the value of **cos(45°)** opens the door to a profound understanding of our universe, driving advancements in fields as diverse as **architecture**, **computer science**, and **engineering**. Below we present a generic diagram for all angles.

Figure-1.

This article will delve into the unique aspects of **cos(45°)**, unfolding its mathematical beauty and real-world significance.

**Definition of cos 45 Degrees**

In **trigonometry**, the **cosine** of an angle in a** right triangle** is defined as the ratio of the length of the **adjacent side** to the length of the **hypotenuse**. When we talk about an angle of **45 degrees**, we’re referring to an angle that’s halfway between** 0** and **90 degrees**. Below we present a generic diagram for the **cosine** **45 degrees**.

Figure-2.

The **cosine** of **45 degrees** holds a special property due to the symmetry of a **45-45-90 triangle**. In this type of triangle, the two non-right angles are **45 degrees**, and the sides opposite these angles are of equal length. Given this equality and the **Pythagorean theorem**, we can conclude that the **cosine** of 45 degrees is equal to **√2 / 2**, or approximately **0.7071**.

It’s important to note that this value is derived based on the **unit circle** definition, often used in trigonometry. In the unit circle, the **cosine** of an angle is the **x-coordinate** of the point on the circle corresponding to that angle. In the case of 45 degrees, or **π/4 radians**, this coordinate is **√2 / 2**. Below we present a generic diagram for the **cosine** **45 degrees **along with the segment of a** cosine function**.

Figure-3.

## Significance of Cos(45°) In Trigonometry

The **cosine** of **45 degrees** is a key value in **trigonometry** due to its unique and elegant properties. Its value, **√2 / 2**, holds a profound place in the study of **mathematics** for several reasons:

**Special Right Triangles**

The **45-45-90 triangle**, also known as an** isosceles right triangle**, is one of the two types of special** right triangles**. Because the two **non-right angles** are both **45 degrees**, and the sides opposite these angles are of equal length, **cos(45°)** is the ratio of the length of either leg to the length of the hypotenuse. Thus, the concept of **cos(45°)** is fundamental in the study of these special right triangles.

**Unit Circle**

In the context of the **unit circle**, which is a central tool in **trigonometry**, the **cosine** of an angle is defined as the** x-coordinate** of the point where the terminal side of the angle intersects the unit circle. For an angle of **45 degrees**, this x-coordinate is **√2 / 2**, once again demonstrating the significance of **cos(45°)**.

**Symmetry**

The value of **cos(45°)** is a beautiful demonstration of the **symmetry** inherent in **trigonometry**. In a **45-45-90 triangle**, the cosine and sine of the **45-degree** angle are equal (**√2 / 2**), highlighting the **complementary** nature of these two fundamental trigonometric functions.

**Complex Numbers and Euler’s Formula**

The **cos(45°)** also plays a crucial role in the field of **complex numbers**, specifically in **Euler’s formula**, which states that e^(ix) = cos(x) + i*sin(x). At **x = 45 degrees** or π/4 radians, both **cosine** and **sine** equal **√2 / 2**, simplifying many calculations in this complex plane.

**Real-World Applications**

Beyond pure **mathematics**, **cos(45°)** is crucial in various **applied** fields, including **physics**, **engineering**, **computer science**, and more. For example, it’s essential in calculating angles of projection for **maximum range**, analyzing waves and oscillations, interpreting rotation matrices in **computer graphics**, and much more.

Hence, the importance of **cos(45°)** in trigonometry and beyond is undeniable. Its mathematical elegance and widespread applicability makes it a cornerstone of mathematical understanding.

**Properties ****of cos 45 Degrees**

The **cosine** of **45 degrees**, denoted as **cos(45°)**, exhibits some intriguing properties in **trigonometry**. Let’s delve into them:

**Value**

The value of **cos(45°)** is **√2/2**, approximately equal to **0.7071**. This is derived from the geometry of the **45-45-90 right triangle** or by using the unit circle definition of the cosine function.

**Symmetry**

One of the fascinating properties of **cos(45°)** is its symmetry with respect to the **sine** function. Specifically, **sin(45°) = cos(45°)**, which is a manifestation of the complementary nature of these two functions. This is due to the fact that in a 45-45-90 triangle, the two legs are of equal length.

**Periodicity**

The cosine function is **periodic** with a period of **360°** or **2π radians**. Therefore, **cos(45° + n*360°) = cos(45°)** for any integer n. This property highlights the repeating nature of the cosine function.

**Even Function**

Cosine is an **even function**, which means that **cos(x) = cos(-x)** for any angle x. Therefore, **cos(45°) = cos(-45°)**, underscoring the function’s symmetry about the y-axis.

**Cosine-Square Identity**

Using the **Pythagorean identity** in trigonometry, we know that **sin²x + cos²x = 1** for any angle x. Substituting **x** with **45°**, we see that the square of **cos(45°)** is **1/2**. This is a direct application of the cosine function to the fundamental identity.

**Double Angle Identity**

The **double angle identity** for cosine can be used to find **cos(90°)** using **cos(45°)**. According to the formula, **cos(2x) = 2cos²x – 1**, therefore, **cos(90°) = 2cos²(45°) – 1 = 0**.

These properties not only **elucidate** the nature of the cosine function and its inherent relationships within **trigonometry** but also facilitate problem-solving across a wide range of **mathematical** contexts.

## Techniques for Constructing and Measuring a Cos(45°) Degree Angle

Constructing and measuring a **45-degree angle**, specifically relating to **cos(45°)**, requires some basic knowledge of geometry. Here are some steps and techniques:

**Constructing a 45-Degree Angle Using a Compass and Straightedge**

**Draw a Base Line**

Using a straightedge, draw a straight line. This line will serve as the **base** or **arm** of your angle.

**Construct a Perpendicular Line**

From a point on this line, use a **compass** to draw an arc intersecting the line. Then, without changing the compass **width**, draw another arc above the line using the **intersection** point as the center. Draw a line through the two **arc** intersections; you now have a **90-degree angle**.

**Bisect the 90-Degree Angle**

To **bisect** an angle means to divide it into two equal parts. Keep the same width on the **compass**, and draw an **arc** cutting both arms at a **90-degree angle**. Draw two more arcs within the angle without changing the width using the **intersections** on the arms as centers. Draw a line from the angle’s **vertex** through the intersection of these two arcs, and you have now bisected the **90-degree** angle into two** 45-degree angles**.

**Measuring a 45-Degree Angle Using a Protractor**

**Position the Protractor**

Place the center of the protractor at the **vertex** of the angle. Ensure the angle’s baseline aligns with the **protractor’s** zero line.

**Read the Measurement**

The second line of the angle will pass through the **45-degree** mark on the protractor if it’s a **45-degree angle**.

**Using Trigonometry**

The **value** of **cos(45°)** can also be confirmed using** trigonometry**. In a **right triangle**, if the lengths of the two legs are equal, the **non-right angles** are both **45 degrees**. If you construct such a triangle and measure the** ratio** of the length of one **leg** to the length of the** hypotenuse**, it should be equal to **cos(45°)**, which is **√2/2**, or approximately **0.7071**.

These techniques are helpful in various fields, from** mathematics** and **physics** to **engineering** and **design**, where the construction and measurement of precise angles are often necessary. Below we present a generic diagram for the **cosine** **45 degrees **constructed in** MATLAB**.

Figure-4.

**Applications **

The value of **cos(45°)** is instrumental in various fields due to its unique and **symmetrical** properties. It plays a pivotal role in several **mathematical** and p**hysical** applications.

**Physics**

In **Projectile Motion**, the maximum horizontal distance (range) a p**rojectile** can travel is achieved when it is launched at an angle of **45 degrees**. This is because the** cosine** and **sine** of **45 degrees** are equal **(cos(45°) = sin(45°))**, maximizing the formula for range, **R = (v² sin(2θ))/g**, where v is the initial velocity,** θ** is the launch angle, and g is the acceleration due to gravity.

**Engineering**

In **Structural Engineering**, the force components are often analyzed in different directions. If a force is applied **diagonally**, the **horizontal** and **vertical** components of that force will be equal when the angle is **45 degrees**, as **cos(45°) = sin(45°)**. This principle is extensively used in analyzing **trusses**, **bridges**, and other **structures**.

**Computer Science**

In **Computer Graphics**, rotation matrices often use **cos(45°)** when rotating objects around the origin by **45 degrees**. Similarly, **cos(45°)** is used in algorithms related to **Image Processing** and **Machine Learning** where rotation of images or data points in **multi-dimensional** space is required.

**Mathematics**

In **Geometry** and **Trigonometry**, **cos(45°)** is essential in solving problems related to right triangles, circles, and more. Its equality with **sin(45°)** is often leveraged to simplify calculations. Also, in **Complex Numbers**, **cos(45°)** appears in Euler’s formula, helping to link trigonometry and **exponential functions**.

**Navigation and GPS**

The concept of **cos(45°)** is used in **GPS Technology** for triangulating positions. When satellites are at **45 degrees** to each other from the viewpoint of the **receiver**, the accuracy of the position fix is **optimized** due to the symmetry that **cos(45°)** provides.

**Telecommunications**

In **Signal Processing**, the Fourier Transform, a method for transforming a signal from the time** domain to** the **frequency domain**, leverages the **cosine** function. Signals can be **decomposed** into a sum of cosine and sine waves, and the **cos(45°)** may appear in calculations involving phase** shifts** or **rotations** in the signal.

Each of these applications underscores the profound impact of the seemingly simple trigonometric value, **cos(45°)**, extending far beyond pure **mathematics**.

**Exercise **

**Example 1**

For the given equation, solve for** x:** **cos(x) = cos(45°)**

### Solution

Given that **cos(45°) = √2/2**, we’re looking for angles x where cos(x) = √2/2.

Since the cosine function is periodic with a period of **360°** or **2π radians**, there are infinite solutions. However, the principal solutions in the range of 0 to 360 degrees are:

x = 45°

and:

x = 360° – 45°

x = 315°

Consequently, the solutions are **x = 45°** and **x = 315°**.

**Example 2**

Find the **length** of the diagonal **d** of a square with side length **s**.

### Solution

In a square, the diagonal splits the square into two 4**5-45-90 triangles**. Therefore, we can use this triangle’s property that the hypotenuse length (the diagonal in this case) is **√2** times the length of a leg (the side of the square).

So, if we let s be the square’s side length, the diagonal d’s length is given by **d = s√2**.

To see how cos(45°) comes into play, recall that in a 45-45-90 triangle, cos(45°) = adjacent/hypotenuse. If we let the side of the square be the adjacent side, then cos(45°) = s/d.

Substituting the known value of cos(45°) = √2/2, we get:

√2/2 = s/d

Solving this equation for d gives us:

d = s√2

as before.

**Example 3**

If a force **F** acts at an angle of **45 degrees** to the horizontal, find the **horizontal component** of the force.

### Solution

The horizontal component Fx of a force F acting at an angle θ to the horizontal is given by:

Fx = Fcos(θ)

If θ = 45°, then we get:

Fx = Fcos(45°)

Given that cos(45°) = √2/2, the horizontal component of the force is:

Fx = F(√2/2)

which simplifies to:

Fx = F√2 / 2

Consequently, the horizontal component of the force is **F√2 / 2**.

*All figures are generated using MATLAB.*