The cosine function (cos) is one of the fundamental trigonometric functions used in mathematics, primarily in the study of angles and triangles. In simple terms, the cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
This function is widely used in geometry, physics, and engineering, making it an essential tool for solving real-world problems. Let’s explore "what is cosine", cos formula, the domain and range of the cosine function, its properties, and its applications in detail focusing on how it fits within trigonometric functions.
What is the Cosine Function?
The cosine function is defined as the ratio of the adjacent side of a right-angled triangle to its hypotenuse.
The cosine function is abbreviated as the cos(x) or cos(θ) where x is the angle in radians and theta θ is the angle in degrees generally. The cosine function can be defined using a unit circle i.e., a circle of unit radius as we will see later in this article. It is periodic in nature and repeats its values after every complete rotation of angles. On a cartesian plane, it can be referred to as the vector component of the hypotenuse parallel to the x-axis.
Cos x = Cos θ = Length of Base/Length of Hypotenuse = b/h
where x is the angle in radians and θ is the equivalent angle in degrees.
Cosine Function Graph
The graph of cosine function resembles the graph of sine function with a basic difference that for x = 0 sin function graph passes from the origin while at x = 0, the cosine function graph passes from (0, 1) at y-aixs. Following is the graph of the value of cosine function i.e. y = cos x
The properties discussed above can be seen in the graph like the periodic nature of the function.
Variation of Cosine Function in Graph
Since the range of cosine function is [-1, 1], therefore it varies from -1 to 1 in the graph. It exhibits its periodic nature as the graph repeats after every length 2π on the x-axis. This reflects that the cosine function has a period of 2π (or 360°).
Cos Value Table
Following table provides the values of cosine function for some common angles in the first quadrant of cartesian plane -
Angle in Degrees (θ) | Angle in Radians (x) | Cos (x) |
|---|
0 | 0 | 1 |
30 | π/6 | √3/2 |
45 | π/4 | 1/√2 |
60 | π/3 | 1/2 |
90 | π/6 | 0 |
We can easily calculate the values of other common angles like 15°, 75°, 195°, -15°, etc. using these values by using the formulas cos (x + y) and cos (x - y) described later in this article.
Check, Trigonometric Table
Cosine Function (Cos) Identities
The basic trigonometric identities related to cosine function is mentioned below:
- sin2(x) + cos2(x) = 1
- cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
- cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
- cos(-x) = cos(x)
- cos(x) = 1/sec(x)
- cos 2x = cos2x - sin2x = 1 - 2sin2x = 2cos2x - 1 = (1 - tan2x/1 + tan2x)
- cos 3x = 4cos3x - 3cos x
Cos in Unit Circle
Cosine Function can be defined using unit circle. Let's understand how we can define cosine function in terms of unit circle.
Consider a line segment OA rotating about the point O where O is the origin of the cartesian plane. Thus, the rotation of OA describes a unit circle (circle of unit radius) centered at the origin O and the point A always lies on this circle. If we drop a perpendicular from A on the x-axis and call the point of intersection as B, and θ is the angle that OA makes with the positive direction of the x-axis, then cos(θ) = projection of hypotenuse on x-axis = OB/|OA| = OB (since |OA| = 1 unit).
Note that the direction OB is important as seen in the following figures. The green segment denotes the length/magnitude and the arrow denote the direction (+ve or -ve) of cos(θ)
Note that the value of cos(θ) is positive for θ belonging to first and fourth quadrant while negative for θ belonging to second and third quadrant.
Domain and Range of Cos Function
We know that for a function, the domain is the permissible input value and range is the output value for that particular input or domain value. Hence, we can assume that function acts like a processor that takes input, processes it, and gives particular output. The domain and range of cos function is discussed below:
- Domain of cosine function: R i.e., set of all real numbers.
- Range of cosine function: [-1, 1], i.e., the output varies between all real numbers between -1 and 1.
Period of a Cosine Function
The function is periodic in nature, i.e., it repeats itself after 2π or 360°. In other words, it repeats itself after every complete rotation. Hence, the period of cosine function is a complete rotation or an angle of 360° (or 2π).
Reciprocal of a Cosine Function
The reciprocal of a cosine function is known as secant function or sec for short. Mathematically, the reciprocal of cosine function is given as
sec(θ) = 1/cos(θ)
As per rules of Reciprocals, if we multiply the Cos x with Sec x the product will be always 1.
Inverse of Cosine Function
The inverse of a cosine function known as arc-cosine function and abbreviated as arccos(x) or cos-1(x) is defined as follows
cos(x) = y
⇒ cos-1(y) = x
Domain and Range of Inverse Cosine Function
The domain and range of Inverse cosine Function are mentioned below:
- Domain of Inverse Cosine Function: All real numbers in range [-1, 1]
- Range of Inverse Cosine Function: All real numbers in range [0, π]
Hyperbolic Cosine Function
Hyperbolic Functions are analog equivalent of Trigonometric Function whose algebraic expression is in the terms of exponential function. The hyperbolic cosine function abbreviated as cosh(x) where x is a hyperbolic angle is a concept of hyperbolic geometry. Like (cos(x), sin(x)) represents a point on a unit circle, (cosh(x), sinh(x)) represents a point on a unit hyperbola i.e., xy = 1 where sinh(x) represents hyperbolic sine function. The algebraic expansion of hyperbolic cos function is given as
cosh(x) = (ex + e-x)/2
More details of hyperbolic functions are beyond the scope of this article, but you can refer to this article.
Cosine Function in Calculus
The branch of calculus in mathematics deals with the differentiation and integration of a given function. Differentiation of function is the rate of change in the function with respect to the independent variable while integration is the reverse process of differentiation that deals with finding the integral of a function whose derivative exist.
Derivative of cosine function
The derivative of cosine function is equal to negative of sine function. Mathematically
d(cos(x))/dx = -sin(x)
Integration of cosine function
The indefinite integral of cosine function is equal to the sine function. Mathematically -
∫cos(x)dx = sin(x) + C, where C is the constant of integration.
Sine and Cosine Functions
Following graph represents the key difference between both sine and cosine function:
Difference between Sine and Cosine Functions
Following table lists the differences between sine and cosine function -
Sine Function (Sin) | Cosine Function (Cos) |
|---|
In a unit circle, sine of an angle is the projection of the hypotenuse on the y-axis. | In a unit circle, cosine of an angle is the projection of the hypotenuse on the x-axis. |
sin(θ) = Height of the right angled triangle / Length of the hypotenuse | cos(θ) = Base of the right angled triangle / Length of the hypotenuse |
Its value is 0 at 0°, 180° and 360°. | Its value is 0 at 90° and 270°. |
Its value is maximum i.e., 1 at 90°. | Its value is maximum i.e., 1 at 0° and 360°. |
Its value is minimum i.e., -1 at 270°. | Its value is minimum i.e., -1 at 180°. |
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Solved Examples on Cosine Function
Here are some solved examples to help you better understand the concept of cosine function.
Example 1: What is the maximum and minimum values of the cosine function?
Solution:
The maximum value of the cosine function is 1 at 0° and 180° while the minimum value of the function is -1 at 180°.
Example 2: At what angle(s) in the range [0, 360] is the value of cosine function 0?
Solution:
The value of cosine function is 0 at the angles 90° and 270°.
Example 3: For what quadrants is the value of cosine function negative?
Solution:
The cosine function is negative in the IInd and IIIrd quadrants.
Example 4: Calculate the value of cos (45°).
Solution:
As per identity 4 given above, cos(-x) = cos(x).
Therefore, cos(-45°) = cos(45°) = 1/√2
Example 5: Calculate the value of cos(15°).
Solution:
Using identity 3 given above -
\cos(15\degree) = \cos(45\degree - 30\degree)
\newline = \cos(45\degree)\cos(30\degree) + \sin(45\degree)\sin(45\degree)
\newline = \frac{1}{\sqrt2}\times\frac{\sqrt3}{2} +
\frac{1}{\sqrt2} \times \frac{1}{2}
\newline = \frac{\sqrt3 + 1}{2\sqrt2}
Example 6: What is cos-1(1/2) in the range [0,π]?
Solution:
Let cos-1(1/2) = y.
Therefore, cos(y) = 1/2 ⇒ y = π/3 in the above given range.
Hence answer is π/3.
Example 7: What is the value of cos(-15°)?
Solution:
Using the identity 3 given above -
\cos(-15\degree)\newline
= \cos(30\degree - 45\degree)\newline
= \cos(30\degree)\cos(45\degree) + \sin(30\degree)\sin(45\degree)\newline
= \frac{\sqrt3}{2}\times\frac{1}{\sqrt{2}} + \frac{1}{2}\times\frac{1}{\sqrt2}\newline
= \frac{\sqrt3 + 1}{2\sqrt2}
.
Alternatively, we can also use the identity cos(-x) = cos(x) and use the value of cos(15°) calculated in example 5.
Example 8: Calculate the area under the graph of cosine function for x = 0 to x = π/2.
Solution:
The given area can be calculated by solving the following definite integral -
\int_0^{\frac{\pi}{2}}\cos(x)dx
\newline = \sin(\frac{\pi}{2}) - \sin(0)
\newline = 1 - 0
\newline = 1
Therefore, answer is 1 unit square.
Example 9: If cos(x) = π/3, find the value of cos(3x) (in decimal form with two decimal digit precision).
Solution:
Using the identity - cos(3x) = 4cos3(x) - 3cos(x) -
cos(3x) = 4⨉(π/3)3-3⨉(π/3) ≅ 4.59 - π = 1.45
Example 10: Find the value of cos(120°).
Solution:
Using the identity for cos(2x)
cos(120°) = cos(2⨉60°) = 1 - 2 sin2(60°) = 1- 2⨉(√3/2)2 = 1 - 3/2 = -1/2
Practice Questions on Cosine Function
Q1. What is the formula to calculate the cos of an angle in a right-angled triangle?
Q2. What is the geometric interpretation of cos on cartesian plane?
Q3. Calculate the value of cos(120°).
Q4. Find the value of cos-1(√3/2) in the range [π, 2π].
Q5. If a pole casts a shadow of same length on the ground, find the angle of the sun with respect to the ground if the sun is in the east direction.
Conclusion
The cosine function is a key component of trigonometric functions, playing an essential role in various fields such as geometry, physics, and engineering. Its relationship with the other trigonometric functions, especially through identities like the Pythagorean identity, makes it a powerful tool in solving mathematical problems. Whether used to model periodic phenomena, calculate angles, or determine distances in right-angled triangles, the cosine function remains fundamental to many practical applications. Understanding its formula, properties, and how it fits into broader trigonometric principles is crucial for anyone studying mathematics or applied sciences.
Cosine Function - FAQs
What is Cosine Function?
The cosine function is one of the fundamental trigonometric functions. It is defined in a right-angled triangle as the ratio of the length of the side adjacent to the concerned angle to the length of the hypotenuse.
Are Cos and Cosine the Same in Trigonometry?
Yes. cos is an abbreviation/short-form of the cosine function.
What is the Range of Cos Function?
The range of the cos or the cosine function is all real numbers ranging from -1 to 1 i.e., [-1,1].
What is the Domain of Cos Function?
The domain of the cos or the cosine function is the ser of all real numbers i.e., R.
What is the Maximum value of Cosine Function?
The maximum value of cosine function is 1 for all angles equivalent to 0° or 360°.
What is the Minimum Value of Cosine Function?
The minimum value of cosine function is -1 for all angles equivalent to 180°.
How to find the Value of Cos(-x)?
The value of cos(-x) can be calculated by calculating the value of cos(x) due to the existence of following identity: cos(-x) = cos(x).
How to Graph Cosine Function?
To draw the graph of cosine function on a cartesian plane, refer to x-axis as representing angles in radians (or degrees) and y-axis as representing the values of cosine function for corresponding angle on x-axis. Now,
- Step 1: Take a subset of x-axis for which you would like to draw the graph.
- Step 2: Divide the x-axis in this range into equidistant points (i.e., there is equal space between all the sub-points). Note the greater the number of divisions, the greater the precision of the resultant graph.
- Step 3: For each of these sub-points x, mark the point (x, cos(x)) on the graph.
- Step 4: Join all the marked points to obtain the graph of cosine function (for the subset of x-axis you selected).
How to find the Period of a Cosine Function?
The period of a cosine function refers to the minimum range of values after which the function starts to repeat itself. We know that the cosine function repeats itself after every complete rotation which means 2π radians. Therefore, the period of cosine function is 2π radians or 360°.
What is Amplitude of a Cosine Function?
The amplitude of a cosine function refers to the maximum displacement of value of the function from the mean position i.e., the x-axis. The amplitude of the cosine function is 1 since the maximum displacement is 1 (for the values -1 and 1 at 180 and 0 degrees respectively. Note, that the range of cosine function is [-amplitude, amplitude].
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