sinx平方/(sinx-cosx)+(sinx+cosx)/(tanx平方-1).化简求值_百度作业帮
拍照搜题，秒出答案
sinx平方/(sinx-cosx)+(sinx+cosx)/(tanx平方-1).化简求值
sinx平方/(sinx-cosx)+(sinx+cosx)/(tanx平方-1).化简求值
解:(sinx+cosx)/[(tanx)^2-1]=(sinx+cosx)/[(sinx/cosx)^2-1]=(cosx)^2(sinx+cosx)/[(sinx)^2-(cosx)^2]
=(cosx)^2/(sinx-cosx)
原式=(sinx)^2/(sinx-cosx)+(sinx+cosx)/[(tanx)^2-1]
=[(sinx)^2+(cosx)^2]/(sinx-cosx)
=1/(sinx-cosx)化简：（1）tanx（cosx-sinx)+(sinx+tanx)/(cotx+cscx) (2)sin^2xtanx+cos^2xcotx+2sinxcosx_百度作业帮
拍照搜题，秒出答案
化简：（1）tanx（cosx-sinx)+(sinx+tanx)/(cotx+cscx) (2)sin^2xtanx+cos^2xcotx+2sinxcosx
化简：（1）tanx（cosx-sinx)+(sinx+tanx)/(cotx+cscx) (2)sin^2xtanx+cos^2xcotx+2sinxcosx
(1)tanx(cosx-sinx)+(sinx+tanx)/(cotx+cscx)=(sinx-sin²x/cosx)+(sinx+sinx/cosx)/(cosx/sinx+1/sinx)=(sinx-sin²x/cosx)+sin²x/cosx=sinx(2)sin²xtanx+cos²xcot+2sinxcosx=sin³x/cosx+cos³x/sinx+2sinxcosx=(sin^4x+cos^4x+2sin²xcos²x)/sinxcosx=(sin²x+cos²x)²/sinxcosx=2/sin2x=2csc2x
1. tanx（cosx-sinx)+(sinx+tanx)/(cotx+cscx) = sinx - tans sinx
+ tanx (cosx+1) / [ cosx+1)/sinx]=
sinx - tans sinx
+ tans sinx =
sinx 2. (sinx)^2 tanx+ (cosx)^2 cotx + 2sinxcosx= (sinx)^2 tanx + sinxcosx +
(cosx)^2 cotx + sinxcosx= tanx + cotxGraphs of the trigonometric functions
LET US BEGIN by introducing some algebraic language. &When we write "n&," where n could be any , we mean "any multiple of&&."
0, &&&, &&2&, &&3&, . &. &.
Problem 1.&&&Which
are indicated by the following, where n could be any integer?
To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload").
The even multiples of &:
0,&&2&, &&4&, &&6&, . &. &.
2n, in algebra, typically signifies an even number.
b) &(2n + 1)&
The odd multiples of &:
&&, &&3&, &&5&, &&7&, . &. &.
+ 1 (or 2n & 1) typically signifies an odd number.
By the zeros of sin &, we mean those values of & for which sin & will equal&0.
Now, where are the zeros of sin &? &That is,
sin & = 0 &when & = ?
We saw in Topic 15 on the
that the value of sin & is equal to the y-coordinate. &Hence, sin & = 0 at & = 0 and &&=&& -- and at all angles
with them.
&In other words,
sin & = 0 &when && = n&.
This will be true, moreover,
for any argument
of the sine function. &For example,
sin 2x = 0 &when the argument 2x = n&;
that is, when
Which numbers are these? &The multiples of&
Problem 2.&&& are the zeros of &y = sin 3x?
At 3x = n&; that is, at
Which numbers are these?
The multiples of&
The zeros of y = sin x are at the multiples of &. &And it is there that the graph crosses the x-axis, because there sin x = 0. &But what is the
maximum value of the graph, and what is its minimum value?
sin x has a maximum value of 1 at&
, and a minimum value of &1
&-- and at all angles
with them.
is the graph of y = sin x:
The height of the curve at every point is the
of the sine.
In the language of functions, y = sin x is an
function. It is symmetrical with respect to the origin.
The independent variable x is the
. &x may be any
. &We may imagine the
unit circle rolled out, in both directions, along the x-axis. &(See Topic 14: &.)
The period
a function
When the values of a function regularly repeat themselves, we say that the function is periodic. &The values of &sin & &regularly repeat themselves
every 2& units.& Hence, sin & is periodic. &Its period is 2&. &(See the previous topic, .)
Definition. &If, for all values of x, the value of a function at x + p is equal to the value at x --
If &f(x + p) = f(x)
-- then we say that the function is periodic and
has period p.
The function &y = sin x &has period 2&, because
sin (x + 2&) = sin x.
The height of the graph at x is equal to the height at x + 2& -- for all&x.
Problem 3.
a) &In the function y = sin x, what is its domain?
a)& (See .)
x may be any real number. && < x < .
b) &What is the range of y = sin x?
sin x has a minimum value of &1, and a maximum of +1.
The graph of y = cos x is the graph of
shifted, or ,
&units to the left.
For, sin (x +&
) &= &cos x. &The student familiar with the sum
formula can easily prove that. (.)
On the other hand, it is possible to see directly that
. &Angle CBD is a right angle.
y = sin ax
Since the graph of &y = sin x &has period 2&, then the constant a in
y = sin ax
indicates the number of periods in an interval of length 2&. &(In y = sin x, a = 1.)
For example,
y = sin 2x
-- that means
there are 2 periods in an interval of length 2&.
If a = 3 --
y = sin 3x
there are 3 periods in that interval:
While if a = &frac12; --
y = sin &frac12;x
there is only half a period in that interval:
The constant a thus signifies how frequently the so many radians per unit of x.
(When the independent variable is the time t, as it often is in physics, then the constant is written as & ("omega"): sin &t. && is called t so many radians per second.)
Problem 4.
a) & For which values of x are the
of y = sin mx?
At mx = n&; that is, at x =&
b) & What is the period of y = sin mx?
. &Since there are m periods in 2&, then one period is 2&
divided by m. Compare the
Problem 5.&&&y = sin 2x.
a) & What does the 2 indicate?
In an interval of length 2&, there are 2 periods.
b) & What is the period of that function?
c)& Where are its zeros?
Problem 6.&&&y = sin 6x.
a) & What does the 6 indicate?
In an interval of length 2&, there are 6 periods.
b) & What is the period of that function?
c)& Where are its zeros?
Problem 7.&&&y = sin &frac14;x.
a) & What does &frac14;
In an interval of length 2&, there is one fourth of a period.
b) & What is the period of that function?
2&/&frac14; = 2&&&4 = 8&.
c)& Where are its zeros?
n&&&frac14;
Here is one period of the graph of y = tan x:
Why is that the graph? &Consider the
DE of tan x in the 4th and 1st quadrants:
As radian x goes from&&
, tan x takes on all real values. That is,
& < tan x < .
Quadrants IV and I constitute
complete period of y = tan x. &In quadrant IV, tan x in quadrant I, and tan 0 = 0.
Again, here is the graph:
,&tan x does not exist. &Therefore
& the lines &x =&&
& and &x =&
&are . (Topic 18 of
Precalculus.)
Here is the complete graph of &y = tan x.
The graph of Quadrants IV and I is repeated in Quadrant II (where tan x is ) and quadrant III (where tan x is positive), and periodically along the entire x-axis.
Problem 7.&&&What is the period of y = tan x?
One period is from&&
.&Hence the period is the
distance between those two points: &.
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