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

Cos Function 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

Related Articles:

• Differentiation of Trigonometric Functions
• Inverse Trigonometric Functions
• Inverse Trig Derivatives

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 II^nd and III^rd 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 –

[Tex]\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} [/Tex]

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 –

[Tex]\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} [/Tex].

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 –

[Tex]\int_0^{\frac{\pi}{2}}\cos(x)dx \newline = \sin(\frac{\pi}{2}) – \sin(0) \newline = 1 – 0 \newline = 1 [/Tex]

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 – 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.

Summary – Cosine Function

The cosine function, denoted as cos(x), is a fundamental trigonometric function defined as the ratio of the base to the hypotenuse in a right-angled triangle and is essential across various fields like physics, engineering, and geometry due to its periodic nature, which is instrumental in modeling wave behaviors. It has a domain of all real numbers and a range from -1 to 1, repeating its cycle every 2π radians or 360 degrees, evident from its wave-like graph that starts at (0,1). In terms of calculus, the derivative of cos(x) is − sin(x), and its integral yields sin(x)+C, with C as the constant of integration. This function also extends to hyperbolic forms, such as cosh(x), enhancing its application in various mathematical contexts and solutions, including wave calculations and oscillations in physical systems.

FAQs on Cosine Function

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].