15 Trig Qs: solving trig eqns, simplying expressions, etc

[KarlyD]

Having lots of trouble - I need some guidance! Please help!

1) The exact value of 6 cos (-61Pi/6) - 2sin(13Pi/4) is...
* (-61Pi/6)x(180 degrees/Pi)= -1830degrees - keep adding 360 degrees and I get 3(root 3)+(root 2)

2) If 2 cos A - 1 = 0, where 0 is less than or equal to A less than 2Pi, then A is equal to...
*I don't understand how to do this at all.

3) The graph of y = tanx intersects the graph of y = -1 at...
*x=(3Pi/4)+nPi, where N is any integer

4) If 9cos^2x+3cosx=0, where 90 degrees < x < 180 degrees, what is the value of x, correct to the nearest tenth?
*I don't understand how to do this at all.

5) Which of the following is an x-intercept of the graph of y=3sin2x?
A) 330 degrees
B) 270 degrees
C) 60 degrees
D) 45 degrees--->> I believe this one is correct.

6) For (csc^2A-1)/(cotAcscA), what is the simplest equivalent trigonometric expression?
*I don't understand how to do this at all.

7) Another form of cos(30 degrees – A)+sin(60 degrees +A) is…
*I got (root 3)cosA+sinA

8) Solve the following for x, where 0 < x < 2Pi.
A) 2 tan x + square root 12 = 0
B) 2 cos^2 x + cos x = 1
*I don't understand how to do this at all.

9) Write the general solutions to the equations above.
*I don't understand how to do this at all.

10) What is the exact value of sin^2(Pi/6)-2sin(Pi/6)cos(Pi/6)+cos^2(-Pi/6)?
*I don't understand how to do this at all.

11) If cosx = -sinx, then sin^2x is…
*I don't understand how to do this at all.

12) For the equation of 4cosx – square root 12 = 0, where 0 is less than x which is less than or equal to 2Pi, determine the measure of x.
*I don't understand how to do this at all.

13) If cos^2x – 6 sin 2 x = 0, one value of x is…
*I don't understand how to do this at all.

14) For the graph of y = 2xin3x, which of the following is not an x-intercept?
A) 60 degrees
B) 90 degrees
C) 240 degrees ---> I believe this one is correct.
D) 2Pi rad

15) For (tanA)/(secA), what is the simplest equivalent trigonometric expression?
*I got sinA but am unsure if it is correct.

 

[stapel]

Having lots of trouble - I need some guidance!

We'll be glad to try to help "guide" you to the solutions, but first we'll need to see where you're lost! :wink:

Please reply showing everything you have tried on each of these fifteen exercises, clearly stating where you are getting stuck.

Thank you!

Eliz.

P.S. Welcome to FreeMathHelp!

 

[soroban]

Re: 15 Trig Qs: solving trig eqns, simplying expressions, et

Hello, KarlyD!

I'll go over the ones you did
. . and you did pretty good!

1) The exact value of: \(\displaystyle \:6\cdot\cos\left(-\frac{61\pi}{6}\right)\,-\,2\cdot\sin\left(\frac{13\pi}{4}\right)\)

* \(\displaystyle \,-\frac{61\pi}{6} \,\times\,\frac{180^o}{\pi}\:=\:-1830^o\)
keep adding 360° and I get: \(\displaystyle \:3\sqrt{3}\,+\,\sqrt{2}\;\) Right!

. . \(\displaystyle \begin{array}{ccccc}\frac{-61\pi}{6}& \,=\,& -1830^o & \,\Rightarrow\, & 330^i \\
\frac{13\pi}{4} & \,=\, & 585^o & \,\Rightarrow\, & 225^o\end{array}\)

We have: \(\displaystyle \:6\cdot\cos(330^o)\,-\,2\cdot\sin(225^o) \:=\:6\left(\frac{\sqrt{3}}{2}\right)\,-\,2\left(-\frac{\sqrt{2}}{2}\right)\:=\:3\sqrt{3}\,+\,\sqrt{2}\)

3) The graph of \(\displaystyle y\:=\:\tan x\) intersects the graph of \(\displaystyle y\,=\,-1\) at:

* \(\displaystyle x\:=\:\frac{3\pi}{4}\,+\,n\pi\), where n is any integer. . Correct!

5) Which of the following is an x-intercept of the graph of \(\displaystyle y\:=\:3\cdot\sin2x\)?
. . \(\displaystyle A)\;330^o\;\;\;B)\;270^o\;\;\;C)\;60^o\;\;\;D)\;45^o\) <---I believe this one is correct. .no

We have: .\(\displaystyle 3\cdot\sin2x\:=\:0\;\;\Rightarrow\;\;\sin2x\:=\:0\)

Then: \(\displaystyle \:2x \:=\:0^o,\,180^o,\,360^o,\,540^o,\,720^o,\,900^o,\,\cdots\)

Hence: \(\displaystyle \:x\:=\:0^o,\,90^o,\,180^o,\,\fbox{270^o},\,360^o,\,450^o,\,\cdots\)

7) Another form of \(\displaystyle \cos(30^o\,-\,A)\,+\,\sin(60^o\,+\,A)\) is:

* I got: \(\displaystyle \:\sqrt{3}\cos A\,+\,\sin A\;\) Yes!

Using the compound-angle identities:

\(\displaystyle (\cos30^o\cos A\,+\,\sin30^o\sin A)\,+\,(\sin60^o\cos A \,+\,\cos60^o\sin A)\)

. . \(\displaystyle = \:\frac{\sqrt{3}}{2}\cos A\,+\,\frac{1}{2}\sin A \,+\,\frac{\sqrt{3}}{2}\cos A\,+\,\frac{1}{2}\sin A \:=\:\sqrt{3}\cos A\,+\,\sin A\)

14) For the graph of \(\displaystyle y \:= \:2sin3x\),
which of the following is not an x-intercept?
A) 60°
B) 90°
C) 240° <--- I believe this one is correct. . no
D) 2Pi radians

We have: \(\displaystyle \:\2\cdot\sin3x\:=\:0\;\;\Rightarrow\;\;\sin3x\:=\:0\)

Then: \(\displaystyle \:3x\:=\:0^o,\,180^o,\,360^o,\,540^o,\,720^o,\,900^o,\,1080^o\)

Hence: \(\displaystyle \:x \:=\:0^o,\,60^o,\,120^o,\,180^o,\,240^o,\,300^o,\,360^o\,(2\pi)\)

\(\displaystyle B)\:90^o\) does not appear among the solutions.

15) For (tanA)/(secA), what is the simplest equivalent trigonometric expression?
* I got sin A. . Correct!

We have: \(\displaystyle \L\:\frac{\tan A}{\sec A} \:=\:\frac{\frac{\sin a}{\sout{\cos A}}}{\frac{1}{\sout{\cos A}}} \:=\:\sin A\)

 

[stapel]

We cannot reasonably provide lessons on missing background material, but we'll be glad to try to find links to you. The following can be viewed as hints (if you are familiar with the content at all) or else as questions (if you aren't familiar, so you're needing links)....

2) If 2 cos A - 1 = 0....
*I don't understand how to do this at all.

Would you be able to solve "2x - 1 = 0"? Are you familiar with the basic reference-angle values for cosine?

4) If 9cos^2x+3cosx=0....
*I don't understand how to do this at all.

Would you be able to solve "9x² + 3x = 0"?

6) For (csc^2A-1)/(cotAcscA), what is the simplest equivalent trigonometric expression?
*I don't understand how to do this at all.

Are you familiar with the definition of cosecant and/or cotangent, in terms of sines and cosines?

8) Solve the following for x, where 0 < x < 2Pi.
A) 2 tan x + square root 12 = 0
B) 2 cos^2 x + cos x = 1
*I don't understand how to do this at all.

A) Do you know how to simplify "sqrt[12]"? Would you be able to solve "2x + sqrt[12] = 0"? Are you familiar with the basic reference-angle values for tangent?

B) Would you be able to solve "2x² + x = 1"?

9) Write the general solutions to the equations above.
*I don't understand how to do this at all.

Are you familiar with the fact that trig functions are periodic (that is, that their graphs and values repeat regularly)?

(You would use this fact to convert the 0-to-2pi solutions from (8) into general solutions, is why I ask. For instance, sin(@) = 1 for pi/2 in the "first" period, and the general solution is for pi/2, plus or minus a period length: pi/2 +/- (2pi)k, for k = 0, 1, 2, 3,....)

10) What is the exact value of sin^2(Pi/6)-2sin(Pi/6)cos(Pi/6)+cos^2(-Pi/6)?
*I don't understand how to do this at all.

Would you be able to factor "x² - 2xy + y²"? Have you learned any sum and/or difference identities for sines and/or cosines?

11) If cosx = -sinx, then sin^2x is…
*I don't understand how to do this at all.

Solve "cos(x) = -sin(x)" (dividing through and applying a basic reference-angle value for tangent might be a good method). Then evaluate sin²(x) for this value of x.

12) For the equation of 4cosx – square root 12 = 0....
*I don't understand how to do this at all.

This one works just like (8-A).

13) If cos^2x – 6 sin 2 x = 0, one value of x is…
*I don't understand how to do this at all.

Convert the squared cosine to a squared-sine expression, using the fundamental Pythagorean Identity, "sin²(@) + cos²(@) = 1". Then solve the resulting quadratic, if you're familiar with how to do that.

Thank you!

Eliz.

 

[KarlyD]

Thank you SO much for your help, everyone. It's been incredibly useful!
=)