Prove that sin210°=
Answers
SOLUTION
hope it helps ☺️
of the original function, and vice versa, as summarized in Figure 1.
A chart that says “Trig Functinos”, “Inverse Trig Functions”, “Domain: Measure of an angle”, “Domain: Ratio”, “Range: Ratio”, and “Range: Measure of an angle”.
Figure 1
For example, if \displaystyle f(x)=\sin xf(x)=sinx, then we would write \displaystyle f^{1}(x)={\sin}^{-1}{x}f
1
(x)=sin
−1
x. Be aware that \displaystyle {\sin}^{-1}xsin
−1
x does not mean \displaystyle \frac{1}{\sin{x}}
sinx
1
. The following examples illustrate the inverse trigonometric functions:
Since \displaystyle \sin\left(\frac{\pi}{6}\right)=\frac{1}{2}sin(
6
π
)=
2
1
, then
π
6
=
sin
−
1
(
1
2
)
.
Since
cos
(
π
)
=
−
1
, then
π
=
cos
−
1
(
−
1
)
.
Since \displaystyle \tan\left(\frac{\pi}{4}\right)=1tan(
4
π
)=1, then
π
4
=
tan
−
1
(
1
)
.
In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if \displaystyle f(a)=bf(a)=b, then an inverse function would satisfy
f
−
1
(
b
)
=
a
.
Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. Figure 2 shows the graph of the sine function limited to
[
−
π
2
,
π
2
]
and the graph of the cosine function limited to [0, π].
Two side-by-side graphs. The first graph, graph A, shows half of a period of the function sine of x. The second graph, graph B, shows half a period of the function cosine of x.
Figure 2. (a) Sine function on a restricted domain of
[
−
π
2
,
π
2
]
; (b) Cosine function on a restricted domain of [0, π]
Figure 3 shows the graph of the tangent function limited to
(
−
π
2
,
π
2
)
.
A graph of one period of tangent of x, from -pi/2 to pi/2.
Figure 3. Tangent function on a restricted domain of
(
−
π
2
,
π
2
)
These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
The inverse sine function
y
=
sin
−
1
x
means \displaystyle x=\sin yx=siny. The inverse sine function is sometimes called the arcsine function, and notated arcsin x.
y
=
sin
−
1
x
has domain [−1, 1] and range
[
−
π
2
,
π
2
]
The inverse cosine function
y
=
cos
−
1
x
means \displaystyle x=\cos yx=cosy. The inverse cosine function is sometimes called the arccosine function, and notated arccos x.
y
=
cos
−
1
x
has domain [−1, 1] and range [0, π]
The inverse tangent function
y
=
tan
−
1
x
means \displaystyle x=\tan yx=tany. The inverse tangent function is sometimes called the arctangent function, and notated arctan x.
y
=
tan
−
1
x
has domain (−∞, ∞) and range
(
−
π
2
,
π
2
)
The graphs of the inverse functions are shown in Figure 4, Figure 5, and Figure 6. Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that
sin
−
1
x
has domain [−1, 1] and range
[
−
π
2
,
π
2
]
,
cos
−
1
x
has domain [−1, 1] and range [0, π], and
tan
−
1
x
has domain of all real numbers and range
(
−
π
2
,
π
2
)
. To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line \displaystyle y=xy=x.
A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions
Figure 4. The sine function and inverse sine (or arcsine) function
A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions.
Figure 5. The cosine function and inverse cosine (or arccosine) function
A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions.
Figure 6. The tangent function and inverse tangent (or arctangent) function
please follow me