I have a test in about an hour and a half and am having some difficulty on the review the teacher gave us to help us prepare. Any help would be great!
*Convert the polar equation to a rectangular equation.
rsin(theta - Pi/4) =5
-The answer is given as y= x+5sq.rt.2
Also, I have a problem on here that says to "use a graphing utility to find the rectangular coordinates of the point to two decimal places."
Polar Coordinates (-2.2, 5pi/9 )
How do I input this in a TI-84 calculator?
For the first question, converting a polar equation to a rectangular equation involves using the trigonometric identities for sine and cosine. Here's how you can do it step by step:
1. Start with the given equation: rsin(theta - Pi/4) = 5.
2. Expand the trigonometric function using the angle difference identity: rsin(theta)cos(-Pi/4) - rcos(theta)sin(-Pi/4) = 5.
3. Simplify the equation using the values of cos(-Pi/4) and sin(-Pi/4). Cos(-Pi/4) equals sqrt(2)/2, and sin(-Pi/4) equals -sqrt(2)/2:
rsin(theta) * sqrt(2)/2 - rcos(theta) * (-sqrt(2)/2) = 5.
4. Rearrange the equation and combine like terms:
(r * sqrt(2)/2) * sin(theta) + (r * sqrt(2)/2) * cos(theta) = 5.
5. Recognize that r * sqrt(2)/2 is equal to (1/sqrt(2)) * r, which simplifies to r/sqrt(2):
(r/sqrt(2)) * sin(theta) + (r/sqrt(2)) * cos(theta) = 5.
6. Substitute y for r * sin(theta) and x for r * cos(theta):
(y/sqrt(2)) + (x/sqrt(2)) = 5.
7. Multiply the equation by sqrt(2) to eliminate the denominators:
y + x = 5 * sqrt(2).
8. Rearrange the equation to have y in terms of x:
y = -x + 5 * sqrt(2).
Therefore, the rectangular equation is y = -x + 5 * sqrt(2).
For the second question about using a TI-84 calculator to find the rectangular coordinates of a point given in polar coordinates, follow these steps:
1. Turn on your TI-84 calculator and press the "MODE" button.
2. Press the right arrow key to select "RADIAN" mode for angle unit.
3. Press the "2nd" button, then press the "." (decimal point) button to access the angle mode options.
4. Select "3: polar" to switch to polar coordinate input mode.
5. Press the "2nd" button, then press the "APPS" button to access the "ANGLE" menu.
6. Press the down arrow key to select "6: >r" to convert the polar angle to radians.
7. Enter the given polar coordinates (-2.2, 5pi/9) by typing "-2.2" and then pressing the comma button ",", followed by typing "5", "pi", "/", "9". It should look like "-2.2, 5pi/9".
8. Press the right arrow key to move the cursor to the "OK" button.
9. Press the "ENTER" button. The calculator will display the rectangular coordinates of the point in the form (x, y).
Make sure to round the coordinates to two decimal places as mentioned in the question.
I hope this helps you with your test preparation!
To convert the polar equation rsin(theta - Pi/4) = 5 to rectangular form, follow these steps:
Step 1: Use the trigonometric identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b) to expand the equation.
rsin(theta)cos(- Pi/4) - rcos(theta)sin(- Pi/4) = 5
Step 2: Simplify the trigonometric functions cos(- Pi/4) and sin(- Pi/4).
rsin(theta)(1/sqrt(2)) + rcos(theta)(1/sqrt(2)) = 5
(1/sqrt(2))rsin(theta) + (1/sqrt(2))rcos(theta) = 5
Step 3: Rearrange the equation and substitute x and y for rcos(theta) and rsin(theta) respectively.
(1/sqrt(2))y + (1/sqrt(2))x = 5
Step 4: Multiply both sides of the equation by sqrt(2) to eliminate the denominators.
sqrt(2)((1/sqrt(2))y) + sqrt(2)((1/sqrt(2))x) = sqrt(2)(5)
y + x = 5sqrt(2)
y = x + 5sqrt(2)
Therefore, the rectangular equation is y = x + 5sqrt(2).
Regarding inputting the polar coordinates (-2.2, 5pi/9) into a TI-84 calculator:
1. Turn on your calculator and press the "MODE" button.
2. Use the arrow keys to highlight "POL" (for polar mode).
3. Press the "2nd" ("MODE") button again to exit the mode menu.
4. Press the "2nd" ("QUIT") button to return to the home screen.
5. Press the "ALPHA" button followed by the letter "Y" ("VARS") button.
6. Use the arrow keys to highlight "1: POLARPOL" and press "ENTER."
7. Enter the value -2.2, press "ENTER," then enter the value 5pi/9, and press "ENTER."
The calculator should now display the rectangular coordinates of the point to two decimal places.