Solving equation with $cos$ and $sin$
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$f(x) = 3cos(x) - 9sin(x)$
Is there an easy way to solve $f(x) = 0$?
I'm drawing a blank. It seems impossible and the solution to the question I'm trying to do skips over showing the solving.
Thanks!
algebra-precalculus trigonometry
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up vote
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favorite
$f(x) = 3cos(x) - 9sin(x)$
Is there an easy way to solve $f(x) = 0$?
I'm drawing a blank. It seems impossible and the solution to the question I'm trying to do skips over showing the solving.
Thanks!
algebra-precalculus trigonometry
1
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
$f(x) = 3cos(x) - 9sin(x)$
Is there an easy way to solve $f(x) = 0$?
I'm drawing a blank. It seems impossible and the solution to the question I'm trying to do skips over showing the solving.
Thanks!
algebra-precalculus trigonometry
$f(x) = 3cos(x) - 9sin(x)$
Is there an easy way to solve $f(x) = 0$?
I'm drawing a blank. It seems impossible and the solution to the question I'm trying to do skips over showing the solving.
Thanks!
algebra-precalculus trigonometry
algebra-precalculus trigonometry
edited Nov 20 at 5:21
Brahadeesh
5,94042058
5,94042058
asked Nov 18 at 18:41
M Do
54
54
1
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43
add a comment |
1
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43
1
1
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43
add a comment |
3 Answers
3
active
oldest
votes
up vote
1
down vote
$f(x)=0$ means $3 cos x = 9 sin x$.
Now if $cos x=0$, then $sin x=1$ or $-1$, so values of $x$ where $cos x=0$ are not a solution to the equation.
Divide by $9 cos x$ on both sides,
$tan x = 1/3$
Now you can probably compute $arctan(1/3)$ by a calculator and $npi + arctan(1/3)$ where $n$ is an integer is the complete set of solutions.
add a comment |
up vote
0
down vote
HINT
We have that
$$f(x)=3cos x - 9 sin x =0 implies 3cos x cdot (1-3 tan x)=0$$
and $cos x=0$ is not a solution.
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
|
show 5 more comments
up vote
0
down vote
$$f(x) = 3cos x-9sin x$$
$$f(x) = 0 implies 0 = 3cos x-9sin x$$
$$implies 9sin x = 3cos x implies tan x = frac{1}{3}$$
$tan x$ takes positive values in the first and third quadrants.
For the first quadrant,
$$x = arctan frac{1}{3}$$
For the third quadrant,
$$x = pi+arctan frac{1}{3}$$
That is the solution for $x in [0, 2pi]$. For a general solution, $tan x$ is periodic every $pi$ radians, so for all $n in mathbb{Z}$,
$$x = pi n+arctan frac{1}{3}$$
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
$f(x)=0$ means $3 cos x = 9 sin x$.
Now if $cos x=0$, then $sin x=1$ or $-1$, so values of $x$ where $cos x=0$ are not a solution to the equation.
Divide by $9 cos x$ on both sides,
$tan x = 1/3$
Now you can probably compute $arctan(1/3)$ by a calculator and $npi + arctan(1/3)$ where $n$ is an integer is the complete set of solutions.
add a comment |
up vote
1
down vote
$f(x)=0$ means $3 cos x = 9 sin x$.
Now if $cos x=0$, then $sin x=1$ or $-1$, so values of $x$ where $cos x=0$ are not a solution to the equation.
Divide by $9 cos x$ on both sides,
$tan x = 1/3$
Now you can probably compute $arctan(1/3)$ by a calculator and $npi + arctan(1/3)$ where $n$ is an integer is the complete set of solutions.
add a comment |
up vote
1
down vote
up vote
1
down vote
$f(x)=0$ means $3 cos x = 9 sin x$.
Now if $cos x=0$, then $sin x=1$ or $-1$, so values of $x$ where $cos x=0$ are not a solution to the equation.
Divide by $9 cos x$ on both sides,
$tan x = 1/3$
Now you can probably compute $arctan(1/3)$ by a calculator and $npi + arctan(1/3)$ where $n$ is an integer is the complete set of solutions.
$f(x)=0$ means $3 cos x = 9 sin x$.
Now if $cos x=0$, then $sin x=1$ or $-1$, so values of $x$ where $cos x=0$ are not a solution to the equation.
Divide by $9 cos x$ on both sides,
$tan x = 1/3$
Now you can probably compute $arctan(1/3)$ by a calculator and $npi + arctan(1/3)$ where $n$ is an integer is the complete set of solutions.
answered Nov 18 at 18:48
Swapnil
580419
580419
add a comment |
add a comment |
up vote
0
down vote
HINT
We have that
$$f(x)=3cos x - 9 sin x =0 implies 3cos x cdot (1-3 tan x)=0$$
and $cos x=0$ is not a solution.
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
|
show 5 more comments
up vote
0
down vote
HINT
We have that
$$f(x)=3cos x - 9 sin x =0 implies 3cos x cdot (1-3 tan x)=0$$
and $cos x=0$ is not a solution.
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
|
show 5 more comments
up vote
0
down vote
up vote
0
down vote
HINT
We have that
$$f(x)=3cos x - 9 sin x =0 implies 3cos x cdot (1-3 tan x)=0$$
and $cos x=0$ is not a solution.
HINT
We have that
$$f(x)=3cos x - 9 sin x =0 implies 3cos x cdot (1-3 tan x)=0$$
and $cos x=0$ is not a solution.
answered Nov 18 at 18:44
gimusi
90.6k74495
90.6k74495
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
|
show 5 more comments
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
I should also mention the interval is [0, 2pi]. So x = pi/2,
– M Do
Nov 18 at 18:46
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
$x=pi/2$ is not a solution. We need that $(1-3tan x)=0$.
– gimusi
Nov 18 at 18:48
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Alright, so the solution is arctan(1/3) and however many solutions I want to include. But... arctan(1/3) is ~18.4, and the solutions are apparently .3217 and 3.464 .... How is this possible? The original equation is actually a derivative and I'm asked to find the points at which the derivative is zero in order to find local extrema.
– M Do
Nov 18 at 18:56
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Your answer is in degrees. The solution is in radians.
– KM101
Nov 18 at 18:58
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
Wow. Of course, thanks.
– M Do
Nov 18 at 18:59
|
show 5 more comments
up vote
0
down vote
$$f(x) = 3cos x-9sin x$$
$$f(x) = 0 implies 0 = 3cos x-9sin x$$
$$implies 9sin x = 3cos x implies tan x = frac{1}{3}$$
$tan x$ takes positive values in the first and third quadrants.
For the first quadrant,
$$x = arctan frac{1}{3}$$
For the third quadrant,
$$x = pi+arctan frac{1}{3}$$
That is the solution for $x in [0, 2pi]$. For a general solution, $tan x$ is periodic every $pi$ radians, so for all $n in mathbb{Z}$,
$$x = pi n+arctan frac{1}{3}$$
add a comment |
up vote
0
down vote
$$f(x) = 3cos x-9sin x$$
$$f(x) = 0 implies 0 = 3cos x-9sin x$$
$$implies 9sin x = 3cos x implies tan x = frac{1}{3}$$
$tan x$ takes positive values in the first and third quadrants.
For the first quadrant,
$$x = arctan frac{1}{3}$$
For the third quadrant,
$$x = pi+arctan frac{1}{3}$$
That is the solution for $x in [0, 2pi]$. For a general solution, $tan x$ is periodic every $pi$ radians, so for all $n in mathbb{Z}$,
$$x = pi n+arctan frac{1}{3}$$
add a comment |
up vote
0
down vote
up vote
0
down vote
$$f(x) = 3cos x-9sin x$$
$$f(x) = 0 implies 0 = 3cos x-9sin x$$
$$implies 9sin x = 3cos x implies tan x = frac{1}{3}$$
$tan x$ takes positive values in the first and third quadrants.
For the first quadrant,
$$x = arctan frac{1}{3}$$
For the third quadrant,
$$x = pi+arctan frac{1}{3}$$
That is the solution for $x in [0, 2pi]$. For a general solution, $tan x$ is periodic every $pi$ radians, so for all $n in mathbb{Z}$,
$$x = pi n+arctan frac{1}{3}$$
$$f(x) = 3cos x-9sin x$$
$$f(x) = 0 implies 0 = 3cos x-9sin x$$
$$implies 9sin x = 3cos x implies tan x = frac{1}{3}$$
$tan x$ takes positive values in the first and third quadrants.
For the first quadrant,
$$x = arctan frac{1}{3}$$
For the third quadrant,
$$x = pi+arctan frac{1}{3}$$
That is the solution for $x in [0, 2pi]$. For a general solution, $tan x$ is periodic every $pi$ radians, so for all $n in mathbb{Z}$,
$$x = pi n+arctan frac{1}{3}$$
answered Nov 18 at 18:53
KM101
3,416417
3,416417
add a comment |
add a comment |
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1
What is the equation? f(x)=0?
– gimusi
Nov 18 at 18:42
Currently the way this is written, it doesn’t have anything that needs solving. What do you mean by “solve”?
– DavidButlerUofA
Nov 18 at 18:43
Yes, solving for x when f(x) = 0
– M Do
Nov 18 at 18:43