The cosine function is a function with R as its domain and [−1, 1] as its range.

**The Cosine Function and Inverse Cosine Function**

The cosine function is a function with **R** as its domain and
[−1, 1] as its range. We write *y *= cos *x *and *y *= cos^{−1} *x *or
*y *= arccos(*x*) to represent the cosine function and
the inverse cosine function, respectively. Since cos ( *x *+ 2*π *) =
cos *x *is true for all real numbers *x *and
cos ( *x *+ *p*) need not be equal to cos *x *for 0 < *p *<
2*π *, *x *∈ **R** , the period
of *y *= cos *x *is 2*π*.

The graph of cosine function is the graph of *y *= cos *x *,
where *x *is a real number. Since cosine function is of period 2*π
*, the graph of cosine function is repeating the same pattern in each of the
intervals ….,
[−4*π *, − 2*π *] , [ − 2*π *, 0] , [0, 2*π *] , [2*π
*, 4*π *] , [4*π *, 6*π *] , …. . Therefore, it
suffices to determine the portion of the graph of cosine function for *x *∈ [0, 2*π *] . We construct the following table to identify
some known coordinate pairs (*x*, *y*) for points on the graph of *y
*= cos *x *, *x *∈ [0, 2*π *].

The table shows that the graph of *y *= cos *x *,
0 ≤ *x *≤ 2*π *, begins at (0,1). As *x *increases from 0
to *π *, the value of *y *= cos *x *decreases from 1 to −1 .
As *x *increases from *π** *to 2* π** *, the value of *y *increases from
−1 to 1. Plot the points listed in the table and connect them with a smooth
curve. The portion of the graph is shown in Fig. 4.10.

The **graph of ***y *= cos *x *, *x *∈ **R** consists of
repetitions of the above portion on either side of the interval [0, 2*π *]
and is shown in Fig. 4.11. From the graph of cosine function, observe that cos *x
*is positive in the first quadrant (for 0 ≤ *x* ≤ π/2), negative in the
second quadrant ( for π/2 < *x *≤ π) and third quadrant ( for π < *x*
< ≤ 3π/2) and again it is positive in the fourth quadrant ( for 3π / 2 < *x*
< 2π).

**Note**

We see from the graph that cos(− *x* ) = cos *x* for
all *x* , which asserts that *y* = cos *x* is an even
function

From the graph of *y *= cos *x *, we observe the
following properties of cosine function:

1.
There is no break or discontinuities in the curve. The cosine
function is continuous.

2.
The cosine function is even, since the graph is symmetric about *y
*-axis.

3.
The maximum value of cosine function is 1 and occurs at *x *=…
, − 2*π *, 0, 2*π *, … and the minimum value is − 1 and occurs at *x
*=… , − *π *, *π *, 3*π *, 5*π *, …. . In other words,
−1≤ cos *x *≤1 for all *x **∈** ***R** .

(i) Shifting the graph of *y* = cos *x* to the right π/2
radians, gives the graph of *y* = cos ( *x* – π/2 ) , which is same
as the graph of y = sin x . Observe that cos ( *x* – π/2 ) = cos ( π/2 - *x*
) = sin *x* .

(ii) y = A sin α x and y = B cos β x always satisfy the
inequalities – |A| ≤ A sin α x ≤ |A| and - |B| ≤ B cos β x ≤ |B| . The
amplitude and period of y = A sin α x are |A| and 2π/|α| , respectively and
those of y = B cos β *x* are |B| and 2π/|β| , respectively.

The functions *y* = A sin α *x* and *y* = B cos β x
are known as **sinusoidal functions**.

(iii) Graphing of y = A sin α x and y = B cos β x are obtained by
extending the portion of the graphs on the intervals [0 , 2π/|α| ] and [0,
2π/|β| ] , respectively.

Phenomena in nature like tides and yearly temperature that cycle
repetitively through time are often modelled using **sinusoids**. For instance, to
model tides using a general form of sinusoidal function *y *= *d *+ *a *cos (*bt
*− *c*) , we give the following steps:

(i) The amplitude of a sinusoidal graph (function) is one-half
of the absolute value of the difference of the maximum and minimum *y *-values
of the graph.

Thus, **Amplitude **, *a *= 1/2 ( max − min) ; Centre line is
*y *= *d *, where *d *= 1/2 ( max + min)

**(ii) Period**, *p *= 2 × ( time from maximum to minimum) ; *b *= 2*π/p*

(iii)* c *= *b *× time at which maximum occurs.

**Model-1**

The depth of water at the end of a dock varies with tides. The
following table shows the depth ( in metres ) of water at various time.

Let us construct a sinusoidal function of the form *y *= *d
*+ *a *cos (*bt *- *c*) to find the depth of water at time *t*.
Here, *a *= 1.4 ; *d *=
2.8 ; *p *= 12 ; *b *= *π/6 *; *c *=
*π /3*.

The required sinusoidal function is *y *= 2.8 +1.4 cos( *π/6
t *– *π/3 *)

The transformations of sine and cosine functions are useful in
numerous applications. A circular motion is always modelled using either the
sine or cosine function.

A point rotates around a circle with centre at origin and radius
4. We can obtain the *y *-coordinate of the point as a function of the
angle of rotation.

For a point on a circle with centre at the origin and radius *a*,
the *y *-coordinate of the point is *y *= *a *sin*θ
*, where *θ *is the angle of rotation. In
this case, we get
the equation *y*(*θ *) = 4 sin*θ *, where *θ *is in radian, the
amplitude is 4 and the period is 2*π *. The amplitude 4 causes a
vertical stretch of the *y *-values of the function sin *θ *by
a factor of 4.

The cosine function is not one-to-one in the entire domain **R**
. However, the cosine function is one-to-one on the restricted domain [0, *π *]
and still, on this restricted domain, the range is [−1, 1]. Now, let us define
the inverse cosine function with [−1, 1]as its domain and with [0, *π *]
as its range.

For -1 ≤ *x *≤ 1, define cos^{-1} *x *as the
unique number *y *in [0, *π*] such that cos *y *= *x
*. In other words, the inverse cosine function cos^{-1} : [-1, 1] → [0, *π*]
is defined by cos^{-1} (*x*) = *y *if and only if cos *y *= *x *and
*y *∈[0, *π *].

**Note**

(i) The sine function is non-negative on the interval [0, *π *],
the range of cos^{−1} *x *. This observation is very important for some of the
trigonometric substitutions in Integral Calculus.

(ii) Whenever we talk about the inverse cosine function, we have
cos *x *: [0, *π *] → [−1, 1] and cos^{−1} *x *: [−1, 1] → [0, *π *] .

(iii) We can also restrict the domain of the cosine function to
any one of the intervals …,[−*π
*, 0],[*π *, 2*π *],…., where it is one-to-one and its range is
[−1, 1].

The restricted domain [0, *π *] is called the **principal domain **of cosine function and
the values of *y *= cos^{−1} *x *, −1 ≤ *x *≤ 1, are known as **principal values **of the function *y *=
cos^{−1} *x *.

From the definition of *y *= cos^{−1} *x *, we observe
the following:

(i)* y *= cos^{-1}*x* if and only if *x *= cos *y* for -1
≤ *x *≤ 1 and 0 ≤ *y *≤ *π *.

(ii) cos (cos^{-1} *x*) = *x *if* |x| *≤
1 and has no sense if |*x*| > 1 .

(iii) cos^{-1} (cos *x*) = *x *if 0 ≤ *x *≤ *π *, the
range of cos^{-1} *x *. Note that cos^{-1} (cos 3*π *) = *π *.

The inverse cosine function cos^{-1} : [-1, 1] →[0,* π *], receives a real number
x in the interval [−1, 1] as an input and gives a real number y in the interval [0, π ]as an output (an
angle in radian measure). Let us
find some points (*x, y*)
using the equation y = cos^{-1} *x* and plot them in the *xy* -plane. Note that the
values of *y* decrease from π to 0 as *x* increases from -1 to 1. The
inverse cosine function is decreasing and continuous in the domain. By
connecting the points by a smooth curve, we get the graph of y = cos^{-1} x as shown in Fig. 4.14

(i) The graph of the function *y *= cos^{-1} *x *is also
obtained from the graph *y *= cos *x *by interchanging *x *and
*y *axes.

(ii) For the function *y *= cos^{-1} *x *, the *x *-intercept
is 1 and the *y *-intercept is π/2 .

(iii) The graph is not symmetric with respect to either origin
or *y *-axis. So, *y *= cos^{-1} *x *is neither even nor odd function.

Find the principal value of cos^{-1} ( √3 / 2 ) .

Let cos^{-1} (√3 / 2 ) = y . Then, cos *y* = √3 / 2.

The **range of the principal values **of *y *= cos^{-1} *x *is [0, *π *].

So, let us find *y *in [0, *π *] such that cos *y* = √3 / 2

But, cos π/6 = √3/2
and π/6 ∈ [0,π ]. Therefore, y = π /6

Thus, the principal value of cos^{-1} (√3/2 ) is π/6 .

Find

It is known that cos^{-1} *x *: [-1, 1] → [0, *π *] is given
by

cos^{-1} *x *= *y *if and only if *x *= cos *y* for -1 ≤ *x *≤ 1
and 0 ≤ *y *≤ *π *.

Thus, we have

**Example 4.7**

Find the domain of cos^{-1} ( [2 + sin *x*] /3 ) .

**Solution**

By definition, the domain of *y *= cos^{-1} *x* is -1 ≤ *x *≤ 1
or |*x| *≤1 . This leads to

-1 ≤ [2 + sin *x*]/*3 *≤ 1 which is same as - 3
≤ 2 + sin *x *≤ 3 .

So, - 5 ≤ sin *x *≤ 1 reduces to -1 ≤ sin *x *≤
1, which gives

- sin^{-1} (1) ≤ *x *≤ sin^{-1} (1) or π/2 ≤ x ≤ π/2 .

Thus, the domain of cos^{-1} ( [2 + sin *x*] /3 ) is [- π/2 , π/2
] .

Tags : Definition, Graph, Properties, Solved Example Problems, Applications , 12th Mathematics : UNIT 4 : Inverse Trigonometric Functions

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12th Mathematics : UNIT 4 : Inverse Trigonometric Functions : The Cosine Function and Inverse Cosine Function | Definition, Graph, Properties, Solved Example Problems, Applications

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