Cosine

Cosine is one of the primary mathematical trigonometric ratios. The ratio of the lengths of the side adjacent to the angle and the hypotenuse of a right-angled triangle is called the cosine function which varies as the angle varies. It is defined in the context of a right-angled triangle for acute angles. The cosine is used to model many real-life scenarios – radio waves, tides, sound waves, musical tones, electrical currents.

The cosine function is simply denoted as cos x, where x is the angle. In this article, we will learn the basic properties of the cosine, its graph, domain and range, derivative, integral, and its power series expansion of cosine. Cos x is a periodic function and has a period of 2π.

Cosine or cos x is a periodic function in trigonometry. Consider a unit circle centered at the origin of the coordinate plane. A variable point P moves on the circumference of this circle. From the figure, we observe that P is in the first quadrant, and OP makes an acute angle of x radians with the positive x-axis. PQ is the perpendicular dropped from P onto the horizontal axis. The triangle is thus formed by joining the points O, P, and Q as shown in the figure, where OQ is the base, and PQ is the height of the triangle.

Hence, the cosine function for the above case can be mathematically written as:

cos x = OQ/OP, Here, x is the acute angle formed between the hypotenuse and the base of a right-angled triangle.

The cosine can be defined as the ratio of the length of the base to the length of the hypotenuse in a right-angled triangle. Mathematically, the cosine function formula in terms of sides of a right-angled triangle is written as:

cos x = Adjacent Side/Hypotenuse = Base/Hypotenuse, where x is the acute angle between the base and the hypotenuse.

As shown in the image above, we note that cos x = OQ/OP = OQ/1 = OQ. As x varies, the value of the cosine varies with the variation in the length of OQ. Now, we will study the variation in the cosine function in the four-quadrant of the coordinate plane.

Case 1: Variation of OQ in the first quadrant.

Suppose that initially, P is on the horizontal axis. Let us consider a movement of P through 90° or π/2 rad. The following figure shows different positions of Q for this movement. Clearly, OQ has decreased in length, from an initial value of 1 (when x is 0 radians) to a final value of 0 (when x is π/2 radians).

Case 2: Variation of OQ in the second quadrant.

Now, we will check the position of P in the second quadrant as we did in the first quadrant and check how the value of the cosine function varies. P subsequently moves from 90° position to 180° position. In this phase of the movement, the length or magnitude of OQ increases, and the value of cosine has decreased from a value of 0 at 90°, to a minimum of -1 at 180°.

Case 3: Variation of OQ in the third quadrant.

When P moves from a position of 180° to a position of 270°, though the length or magnitude of OQ decreases. But since the direction is along the negative y-axis, the actual value of cos x increases from -1 to 0. Thus, the value of the cosine for angle x increases.

Case 4: Variation of OQ in the fourth quadrant.

Finally, when P moves from a position of 270° to a position of 360°, OQ increases from 0 to 1 (once again). The length or magnitude of OQ increases along with the algebraic value of OQ increasing. Thus, the value of the cosine function for angle x increases.

We can now plot this variation on a graph. The horizontal axis represents the input variable x as the angle in radians, and the vertical axis represents the value of the cosine function for x. Merging the response of variation in the value of PQ for all four quadrants, we obtained the complete plot of cos x vs x, for one complete cycle of 0 radians to 2π radians (0° to 360°). The plot thus obtained is shown below:

We study the value of the cosine function for some specific angles as they are easy to remember. These cosine values are used in solving different mathematical problems. Some of these values of the cosine are listed below in the trigonometric table:

Properties of cosine depend upon the quadrant in which the angle lies. The cosine function is a special trigonometric function and has many properties. Some of them are listed below:

• The cos x graph repeats itself after 2π, which suggests the function is periodic with a period of 2π.
• Cos x is an even function because cos(−x) = cos x.
• The domain of cosine function is all real numbers and the range is [-1,1].
• The reciprocal of the cosine function is the secant function.
• Power series expansion of the cosine is cos x = \(\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n}}{(2n)!}\)

In trigonometry, there are several identities involving the cosine function. These identities are very useful in solving various math problems. Some of them are listed below:

• cos x = 1/ sec x
• Inverse of cosine function = cos^-1x = arccos x, where x lies in [-1, 1]
• sin²x + cos²x = 1
• cos (x + y) = cos x cos y - sin x sin y
• cos (x - y) = cos x cos y + sin x sin y
• cos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x
• Derivative of cos x: d(cos x)/dx = -sin x
• Integral of cosine function: ∫cos x dx = sin x + C, where C is the constant of integration.

Related Topics

• Sine Function
• Inverse Trigonometric Ratios
• Trigonometric Table
• Trigonometric Ratios

Important Notes on Cosine Function

• Cosine Function can be mathematically written as:
  cos x = Adjacent Side/Hypotenuse = Base/Hypotenuse
• Cosine Function is a periodic function with a period of 2π.
• The domain of cos x is (−∞, ∞) and the range is [−1,1].

What is Cosine in Trigonometry?

The cosine of an angle is a trigonometric function. The ratio of the lengths of the side adjacent to the angle and the hypotenuse of a right-angled triangle is called the cosine function which varies as the angle varies. It is generally denoted by cos x, where x is the angle between the base and hypotenuse. 

What are the Properties of the Cosine?

Some of the properties of the cosine function are:

• The cosine graph repeats itself after 2π, which suggests the function is periodic with a period of 2π.
• The cosine function is an even function because cos(−x) = cos x.
• The domain of cos x is all real numbers and the range is [-1,1].
• The reciprocal of the cos x is sec x.
• Power series expansion of the cosine function is cos x = \(\sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n}}{(2n)!}\)

What is the Meaning of Cosine?

The cosine can be defined as the ratio of the length of the base to the length of the hypotenuse in a right-angled triangle. It gives the value of the cosine function for angle x, denoted by cos x.

What is the Inverse Trigonometric Function of Cosine Function?

Inverse of cosine function = cos^-1x = arccos x, where x lies in [-1, 1]. It is the inverse function of the cosine and is pronounced as 'inverse cosine' or 'arc cosine'.

How do you Write a Cosine Function?

A cosine function can be written as cos x = Adjacent Side/Hypotenuse = Base/Hypotenuse

What does the Cosine Graph Look Like?

The cosine function curve is an up-down curve that repeats after every 2π radians.

What is the Period of Cosine Function?

A period of a function is when the function has a specific horizontal shift, P, which results in a function equal to the original function, i.e., f(x+P) = f(x) for all values of x within the domain of f. The period of the cosine function is 2π.

Is the Cosine Function Even or Odd?

A function f(x) is an even function if f(-x) = f(x), for all x and it is odd if f(-x) = -f(x), for all x. The cosine function is an even function because cos(−x) = cos x.

What is the Ratio for Cosine?

The ratio for cosine is given by the ratio of the length of the base of a right-angled triangle and the length of the hypotenuse.