Answer:
B- doubling a cube
D- trisecting any angle
Step-by-step explanation:
APEX
Answer:
![y=-2\,(x+3)^2-3](https://tex.z-dn.net/?f=y%3D-2%5C%2C%28x%2B3%29%5E2-3)
Step-by-step explanation:
Notice that they are asking you to write the equation of the parabola in vertex form, that is using the coordinates of the vertex
in the expression:
![y-y_v=a\,(x-x_v)^2\\y=a\,(x-x_v)^2+y_v](https://tex.z-dn.net/?f=y-y_v%3Da%5C%2C%28x-x_v%29%5E2%5C%5Cy%3Da%5C%2C%28x-x_v%29%5E2%2By_v)
we can start by directly replacing the given vertex coordinates (-3, -3) in the expression, and then using the extra info on the point the parabola goes through in order to find the parameter
:
![y=a\,(x-x_v)^2+y_v\\y=a\,(x+3)^2+(-3)\\y=a\,(x+3)^2-3\\-5=a\,(-2+3)^2-3\\-5=a\,(1)-3\\a=-5+3\\a=-2](https://tex.z-dn.net/?f=y%3Da%5C%2C%28x-x_v%29%5E2%2By_v%5C%5Cy%3Da%5C%2C%28x%2B3%29%5E2%2B%28-3%29%5C%5Cy%3Da%5C%2C%28x%2B3%29%5E2-3%5C%5C-5%3Da%5C%2C%28-2%2B3%29%5E2-3%5C%5C-5%3Da%5C%2C%281%29-3%5C%5Ca%3D-5%2B3%5C%5Ca%3D-2)
So, now we can write the full expression for the parabola:
![y=a\,(x+3)^2-3\\y=-2\,(x+3)^2-3](https://tex.z-dn.net/?f=y%3Da%5C%2C%28x%2B3%29%5E2-3%5C%5Cy%3D-2%5C%2C%28x%2B3%29%5E2-3)
Solution:
Given:
![240^0](https://tex.z-dn.net/?f=240%5E0)
To get sin 240 degrees:
240 degrees falls in the third quadrant.
In the third quadrant, only tangent is positive. Hence, sin 240 will be negative.
![sin240^0=sin(180+60)](https://tex.z-dn.net/?f=sin240%5E0%3Dsin%28180%2B60%29)
Using the trigonometric identity;
![sin(x+y)=sinx\text{ }cosy+cosx\text{ }siny](https://tex.z-dn.net/?f=sin%28x%2By%29%3Dsinx%5Ctext%7B%20%7Dcosy%2Bcosx%5Ctext%7B%20%7Dsiny)
Hence,
![\begin{gathered} sin(180+60)=sin180cos60+cos180sin60 \\ sin180=0 \\ cos60=\frac{1}{2} \\ cos180=-1 \\ sin60=\frac{\sqrt{3}}{2} \\ \\ Thus, \\ sin180cos60+cos180sin60=0(\frac{1}{2})+(-1)(\frac{\sqrt{3}}{2}) \\ sin180cos60+cos180sin60=0-\frac{\sqrt{3}}{2} \\ sin180cos60+cos180sin60=-\frac{\sqrt{3}}{2} \\ \\ Hence, \\ sin240^0=-\frac{\sqrt{3}}{2} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20sin%28180%2B60%29%3Dsin180cos60%2Bcos180sin60%20%5C%5C%20sin180%3D0%20%5C%5C%20cos60%3D%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20cos180%3D-1%20%5C%5C%20sin60%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Thus%2C%20%5C%5C%20sin180cos60%2Bcos180sin60%3D0%28%5Cfrac%7B1%7D%7B2%7D%29%2B%28-1%29%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%29%20%5C%5C%20sin180cos60%2Bcos180sin60%3D0-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5C%5C%20sin180cos60%2Bcos180sin60%3D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Hence%2C%20%5C%5C%20sin240%5E0%3D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5Cend%7Bgathered%7D)
To get cos 240 degrees:
240 degrees falls in the third quadrant.
In the third quadrant, only tangent is positive. Hence, cos 240 will be negative.
![cos240^0=cos(180+60)](https://tex.z-dn.net/?f=cos240%5E0%3Dcos%28180%2B60%29)
Using the trigonometric identity;
![cos(x+y)=cosx\text{ }cosy-sinx\text{ }siny](https://tex.z-dn.net/?f=cos%28x%2By%29%3Dcosx%5Ctext%7B%20%7Dcosy-sinx%5Ctext%7B%20%7Dsiny)
Hence,
![\begin{gathered} cos(180+60)=cos180cos60-sin180sin60 \\ sin180=0 \\ cos60=\frac{1}{2} \\ cos180=-1 \\ sin60=\frac{\sqrt{3}}{2} \\ \\ Thus, \\ cos180cos60-sin180sin60=-1(\frac{1}{2})-0(\frac{\sqrt{3}}{2}) \\ cos180cos60-sin180sin60=-\frac{1}{2}-0 \\ cos180cos60-sin180sin60=-\frac{1}{2} \\ \\ Hence, \\ cos240^0=-\frac{1}{2} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20cos%28180%2B60%29%3Dcos180cos60-sin180sin60%20%5C%5C%20sin180%3D0%20%5C%5C%20cos60%3D%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20cos180%3D-1%20%5C%5C%20sin60%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Thus%2C%20%5C%5C%20cos180cos60-sin180sin60%3D-1%28%5Cfrac%7B1%7D%7B2%7D%29-0%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%29%20%5C%5C%20cos180cos60-sin180sin60%3D-%5Cfrac%7B1%7D%7B2%7D-0%20%5C%5C%20cos180cos60-sin180sin60%3D-%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Hence%2C%20%5C%5C%20cos240%5E0%3D-%5Cfrac%7B1%7D%7B2%7D%20%5Cend%7Bgathered%7D)
To get tan 240 degrees:
240 degrees falls in the third quadrant.
In the third quadrant, only tangent is positive. Hence, tan 240 will be positive.
![tan240^0=tan(180+60)](https://tex.z-dn.net/?f=tan240%5E0%3Dtan%28180%2B60%29)
Using the trigonometric identity;
![tan(180+x)=tan\text{ }x](https://tex.z-dn.net/?f=tan%28180%2Bx%29%3Dtan%5Ctext%7B%20%7Dx)
Hence,
![\begin{gathered} tan(180+60)=tan60 \\ tan60=\sqrt{3} \\ \\ Hence, \\ tan240^0=\sqrt{3} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20tan%28180%2B60%29%3Dtan60%20%5C%5C%20tan60%3D%5Csqrt%7B3%7D%20%5C%5C%20%20%5C%5C%20Hence%2C%20%5C%5C%20tan240%5E0%3D%5Csqrt%7B3%7D%20%5Cend%7Bgathered%7D)
To get cosec 240 degrees:
![\begin{gathered} cosec\text{ }x=\frac{1}{sinx} \\ csc240=\frac{1}{sin240} \\ sin240=-\frac{\sqrt{3}}{2} \\ \\ Hence, \\ csc240=\frac{1}{\frac{-\sqrt{3}}{2}} \\ csc240=-\frac{2}{\sqrt{3}} \\ \\ Rationalizing\text{ the denominator;} \\ csc240=-\frac{2}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}} \\ \\ Thus, \\ csc240^0=-\frac{2\sqrt{3}}{3} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20cosec%5Ctext%7B%20%7Dx%3D%5Cfrac%7B1%7D%7Bsinx%7D%20%5C%5C%20csc240%3D%5Cfrac%7B1%7D%7Bsin240%7D%20%5C%5C%20sin240%3D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Hence%2C%20%5C%5C%20csc240%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B-%5Csqrt%7B3%7D%7D%7B2%7D%7D%20%5C%5C%20csc240%3D-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D%20%5C%5C%20%20%5C%5C%20Rationalizing%5Ctext%7B%20the%20denominator%3B%7D%20%5C%5C%20csc240%3D-%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D%5Ctimes%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B%5Csqrt%7B3%7D%7D%20%5C%5C%20%20%5C%5C%20Thus%2C%20%5C%5C%20csc240%5E0%3D-%5Cfrac%7B2%5Csqrt%7B3%7D%7D%7B3%7D%20%5Cend%7Bgathered%7D)
To get sec 240 degrees:
![\begin{gathered} sec\text{ }x=\frac{1}{cosx} \\ sec240=\frac{1}{cos240} \\ cos240=-\frac{1}{2} \\ \\ Hence, \\ sec240=\frac{1}{\frac{-1}{2}} \\ sec240=-2 \\ \\ Thus, \\ sec240^0=-2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20sec%5Ctext%7B%20%7Dx%3D%5Cfrac%7B1%7D%7Bcosx%7D%20%5C%5C%20sec240%3D%5Cfrac%7B1%7D%7Bcos240%7D%20%5C%5C%20cos240%3D-%5Cfrac%7B1%7D%7B2%7D%20%5C%5C%20%20%5C%5C%20Hence%2C%20%5C%5C%20sec240%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B-1%7D%7B2%7D%7D%20%5C%5C%20sec240%3D-2%20%5C%5C%20%20%5C%5C%20Thus%2C%20%5C%5C%20sec240%5E0%3D-2%20%5Cend%7Bgathered%7D)
To get cot 240 degrees:
Answer A would the one that's false