COS

Returns the cosine of the given angle (in radians).

Syntax:

COS(angle)

returns the (trigonometric) cosine of angle, the angle in radians.To return the cosine of an angle in degrees, use the RADIANS function.

Example:

COS(PI()/2)

returns 0, the cosine of PI/2 radians

COS(RADIANS(60))

returns 0.5, the cosine of 60 degrees

Application:

One application of the cosine function is modeling the voltage of an alternating current (AC) circuit.

In an AC circuit, the voltage is not constant; it oscillates over time. This oscillation can be modeled by a sinusoidal function, such as the cosine function, because the voltage at any given time is dependent on the phase of the AC cycle.

The formula for the voltage in a simple AC circuit can be expressed as:

V(t)=Vpeak⋅cos(ωt)

Where:

V(t) is the voltage at time t
Vpeak is the maximum (peak) voltage
ω is the angular frequency (in radians per second)
t is the time (in seconds)

Let's consider a specific example: a household power outlet with a peak voltage of 170 V and a frequency of 60 Hz.

First, we need to convert the frequency (f) from Hz to angular frequency (ω) in rad/s:

ω=2πf=2π⋅60≈377 rad/s

So, the equation for the voltage would be:

V(t)=170⋅cos(377t)

This equation allows us to calculate the voltage at any given time t. Let's create a table to show the voltage at different points in the cycle:

	Time (t) in seconds	Angle (377t) in radians	cos(377t)	Voltage (V(t)) in volts
	A	B	C	D
1	0	0	1	170
2	0.004166667	1.570833333	0	0
3	0.008333333	3.141666667	-1	-170
4	0.0125	4.7125	0	0
5	0.016666667	6.283333333	1	170

As you can see from the table, the cosine function accurately models the oscillation of the voltage. The voltage starts at its peak, decreases to zero, goes to a negative peak, returns to zero, and then completes the cycle by returning to the positive peak. This cyclical behavior is a fundamental characteristic of the cosine function.