Question

1. convert ( 290^{circ} ) to radians.

a. ( \frac{17pi}{9} )
b. ( \frac{29pi}{18} )
c. ( \frac{16pi}{9} )
d. ( \frac{31pi}{18} )

1. identify the center and the radius of the following circle:

( (x - 5)^2+(y + 2)^2 = 64 )
a. center: ( (-5,2) ) radius ( = 8 )
b. center: ( (-5,-2) ) radius ( = 8 )
c. center: ( (5,-2) ) radius ( = 8 )
d. center: ( (5,2) ) radius ( = 8 )

1. solve for the missing angle.

a. ( 145^{circ} )
b. ( 124^{circ} )
c. ( 98^{circ} )
d. ( 41^{circ} )

1. solve for the missing angle.

a. ( 64^{circ} )
b. ( 136^{circ} )
c. ( 85^{circ} )
d. ( 100^{circ} )

1. find the missing angle.

a. ( 69^{circ} )
b. ( 70^{circ} )
c. ( 39^{circ} )
d. ( 42^{circ} )

1. find the missing angle.

a. ( 178^{circ} )
b. ( 93^{circ} )
c. ( 123^{circ} )
d. ( 144^{circ} )

1. what is the missing value in the unit circle?

a. ( 300^{circ} )
b. ( 315^{circ} )
c. ( 275^{circ} )
d. ( 360^{circ} )

Explanation:

1. Convert \(290^{\circ}\) to radians

Step1: Use the conversion formula

To convert degrees to radians, use the formula \(x^{\circ}=x\times\frac{\pi}{180}\) radians.
For \(x = 290\), we have \(290\times\frac{\pi}{180}=\frac{290\pi}{180}=\frac{29\pi}{18}\) (simplify by dividing numerator and denominator by 10).

2. Identify the center and radius of the circle \((x - 5)^{2}+(y + 2)^{2}=64\)

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Compare with the given equation

For \((x - 5)^{2}+(y+2)^{2}=64=(8)^{2}\), we have \(h = 5\), \(k=-2\) and \(r = 8\).

3. Solve for the missing angle in the first circle - the sum of angles around a point is \(360^{\circ}\)

Step1: Set up the equation

Let the missing angle be \(y\). Then \(110 + 50+145+y=360\)

Step2: Simplify the left - hand side

\(110+50 + 145+y=305 + y\)

Step3: Solve for \(y\)

\(y=360 - 305=41^{\circ}\)

4. Solve for the missing angle in the second circle - the sum of angles around a point is \(360^{\circ}\)

Step1: Set up the equation

Let the missing angle be \(z\). Then \(85+89 + 136+z=360\)

Step2: Simplify the left - hand side

\(85+89+136+z=310+z\)

Step3: Solve for \(z\)

\(z=360 - 310 = 64^{\circ}\)

5. Find the missing angle in the inscribed - angle related circle (the measure of an inscribed angle is half the measure of its intercepted arc)

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle \(\angle E\) is half the measure of the arc \(FG\). The arc \(FG\) is \(140^{\circ}\), so \(\angle E=\frac{1}{2}\times140^{\circ}=70^{\circ}\)

6. Find the missing angle in the cyclic quadrilateral (the sum of opposite angles in a cyclic quadrilateral is \(180^{\circ}\))

Step1: Recall the cyclic - quadrilateral property

If \(\angle Q = 87^{\circ}\), and \(\angle M\) is the missing angle. In a cyclic quadrilateral \(LMNO\), \(\angle Q+\angle M=180^{\circ}\)

Step2: Solve for \(\angle M\)

\(\angle M=180 - 87=123^{\circ}\)

7. Find the missing value in the unit circle

Step1: Recall the coordinates on the unit circle

The coordinates \((\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\) correspond to an angle. Using the reference - angle and quadrant knowledge.
The angle with \(\cos\theta=\frac{\sqrt{2}}{2}\) and \(\sin\theta=-\frac{\sqrt{2}}{2}\) is in the fourth quadrant. The reference angle is \(45^{\circ}\), and the angle \(\theta = 360 - 45=315^{\circ}\)

Answer:

1. d. $\frac{31\pi}{18}$
2. c. Center:$(5, - 2)$ Radius = 8
3. d. $41^{\circ}$
4. a. $64^{\circ}$
5. b. $70^{\circ}$
6. c. $123^{\circ}$
7. b. $315^{\circ}$