the general solution of cosx - sinx equal to 1
Answers
Answer:
x
=
2
n
π
and
x
=
(
4
n
+
1
)
π
2
,
n
=
0
,
±
1
,
±
2
,
±
3
.
...
Explanation:
The given equation is equivalent to
1
√
2
sin
x
+
1
√
2
cos
x
=
1
√
2
.
This can be written as
cos
(
x
−
π
4
)
=
cos
(
π
4
)
The general solution of this equation ls
x
−
π
4
=
2
n
π
±
π
4
,
n
=
0
,
±
1
,
±
2
,
...
,
So,
x
=
2
n
π
and
x
=
(
4
n
+
1
)
π
2
,
n
=
0
,
±
1
,
±
2
,
±
3
.
...
Having noted that there were 40K viewers for the answers by me,
Hero and Nghi, I think I could invoke more interest by including the
solutions for
cos
x
−
sin
x
=
1
, and for that matter,
sec
x
±
tan
x
=
1
, that become
cos
x
−
sin
x
=
1
and
cos
x
+
sin
x
=
1
, upon multiplication by
cos x, when
x
≠
an odd multiple of
π
2
.
For cos x - sin x = 1,
the general solution is
x
=
2
n
π
and
x
=
(
4
n
−
1
)
π
2
,
n
=
0
,
±
1
,
±
2
,
±
3
.
...
Note the change in the multiple from
(
4
n
+
1
)
→
(
4
n
−
1
)
.
For
sec
x
±
tan
x
=
1
, it is same sans
(
4
n
±
1
)
π
2
. It is just
x
=
2
n
π
See x-intercepts as graphical solutions.
Graph on uniform scale for solutions
x
=
...
−
2
π
,
−
3
π
2
,
0
,
π
2
,
2
π
.
.
of cos x + sin x = 1:
graph{y-cos x - sin x +1 = 0[-7 7 -3 4]}
Graph on uniform scale for solutions
x
=
−
2
π
,
−
π
2
,
0
,
3
π
2
,
2
π
,
.
.
of
cos
x
−
sin
x
=
1
:
graph{y-cos x + sin x +1 = 0[-7 7 -3 4]}
See combined graph for solutions
0
,
±
2
π
,
±
4
π
,
...
of
sec
x
±
tan
x
=
1
:
graph{(y- sec x - tan x +1)(y- sec x+ tan x +1)=0[-13 13 -6.5 6.5]}
Answer:
cos (X) -sin (X) =1
Divide by √2 on both sides
1/√2 cosx-1/√2 sinx=1/√2
therefor ,
➡ cos (π/4) cosx - sin(π/ 4)sinx=cos(π/4)
➡cos(x+π/4)=cos(π/4)
now, costhita =cosalpha implies,
thita =2nπ+ alpha, n€Z
➡x+π/4=2nπ+π/4
➡x=2nπ+π/4- π/4
➡x=2nπ+π/4-π/4 or
X = 2nπ-π/4-π/4