Trig help

[Jordannn1990]

trig help(done)

Hello, I am a sophmore in college and need help with a few trig problems.

1. Find all degree solutions to the following equation. cos( a-40) = sqrt3/2

2. Solve for all radian solutions sin(x) -2sin(x)cos(x)=0

and 3. cos(a+pi/12) = -1/sqrt2 Find all radian solutions to the following equation

I have no idea how to do numbers 1 and 3.

 

[Deleted member 4993]

Hello, I am a sophmore in college and need help with a few trig problems on a test review sheet.

1. Find all degree solutions to the following equation. cos( a-40) = sqrt3/2

2. Solve for all radian solutions sin(x) -2sin(x)cos(x)=0

I have 0, pi/3 and need the last angle.

and 3. cos(a+pi/12) = -1/sqrt2 Find all radian solutions to the following equation

I have no idea how to do numbers 1 and 3.

Please share your work for #2 ( #1 and #3 re actually easier than #2).

Also, you would need to restrict the domain - to have finite number of solutions.

 

[Jordannn1990]

Please share your work for #2 ( #1 and #3 re actually easier than #2).

Sinx=0 = 90 , 180. 2pi, pi.


cosx= 1/2 = 60, 240. pi/3, 4pi/3

 

[Deleted member 4993]

Sinx=0 = 90 , 180. 2pi, pi.


cosx= 1/2 = 60, 240. pi/3, 4pi/3

Sin (90) is not equal to 0.

Show complete work - how did you get to these conditions [cosx= 1/2 = 60, 240.] from the given problem?

 

[soroban]

Hello, Jordannn1990!

I will solve for the principal values.
You can generalize them if needed.

1. Find all degree solutions: .\(\displaystyle \cos(\theta-40) \:=\:\frac{\sqrt3}{2}\)

\(\displaystyle \cos(\theta - 40^o) \:=\:\frac{\sqrt{3}}{2}\)

n . . \(\displaystyle \theta - 40^o \:=\:\pm 30^o\)

. . . . . . . . \(\displaystyle \theta \:=\:40^o \pm 30^o \:=\:\begin{Bmatrix}70^o \\ 10^o\end{Bmatrix}\)

2. Find all radian solutions: .\(\displaystyle \sin x -2\sin x\cos x \:=\:0\)

Factor: .\(\displaystyle \sin x(1-2\cos x) \:=\:0\)

We have two equations to solve:

.\(\displaystyle \sin x \:=\:0 \quad\Rightarrow\quad x \:=\:0,\,\pi\)

. . \(\displaystyle 1-2\cos x \:=\:0 \quad\Rightarrow\quad \cos x \:=\:\frac{1}{2} \quad\Rightarrow\quad x \:=\:\pm\frac{\pi}{3}\)

3. Find all radian solutions: .\(\displaystyle \cos\left(\theta+\tfrac{\pi}{12}\right) \:=\: \text{-}\frac{1}{\sqrt2}\)

\(\displaystyle \cos\left(\theta + \tfrac{\pi}{12}\right) \:=\:-\frac{1}{\sqrt{2}}\)

. . . . \(\displaystyle \theta + \tfrac{\pi}{12} \:=\:\begin{Bmatrix}\frac{3\pi}{4} \\ \frac{5\pi}{4} \end{Bmatrix}\)

. . . . . . . .\(\displaystyle \theta \:=\:\begin{Bmatrix}\frac{3\pi}{4} - \frac{\pi}{12} &=& \frac{2\pi}{3} \\ \frac{5\pi}{4} - \frac{\pi}{12} &=& \frac{7\pi}{6} \end{Bmatrix}\)

 

[Jordannn1990]

Hello, Jordannn1990!

I will solve for the principal values.
You can generalize them if needed.


\(\displaystyle \cos(\theta - 40^o) \:=\:\frac{\sqrt{3}}{2}\)

n . . \(\displaystyle \theta - 40^o \:=\:\pm 30^o\)

. . . . . . . . \(\displaystyle \theta \:=\:40^o \pm 30^o \:=\:\begin{Bmatrix}70^o \\ 10^o\end{Bmatrix}\)




Factor: .\(\displaystyle \sin x(1-2\cos x) \:=\:0\)

We have two equations to solve:

.\(\displaystyle \sin x \:=\:0 \quad\Rightarrow\quad x \:=\:0,\,\pi\)

. . \(\displaystyle 1-2\cos x \:=\:0 \quad\Rightarrow\quad \cos x \:=\:\frac{1}{2} \quad\Rightarrow\quad x \:=\:\pm\frac{\pi}{3}\)




\(\displaystyle \cos\left(\theta + \tfrac{\pi}{12}\right) \:=\:-\frac{1}{\sqrt{2}}\)

. . . . \(\displaystyle \theta + \tfrac{\pi}{12} \:=\:\begin{Bmatrix}\frac{3\pi}{4} \\ \frac{5\pi}{4} \end{Bmatrix}\)

. . . . . . . .\(\displaystyle \theta \:=\:\begin{Bmatrix}\frac{3\pi}{4} - \frac{\pi}{12} &=& \frac{2\pi}{3} \\ \frac{5\pi}{4} - \frac{\pi}{12} &=& \frac{7\pi}{6} \end{Bmatrix}\)

Thanks alot. I appreciate it. Now I can finish my review with everything correct.

 

[Jordannn1990]

Bump added more problems I need help with.