Question

the measure of an angle in standard position is given. find two positive angles and two negative angles (in degrees) that are coterminal with the 225°. the measure of an angle in standard position is given. find two positive angles and two negative angles (in radians) that are coterminal with the -\frac{5\pi}{4} rad

Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles of an angle $\theta$ can be found by adding or subtracting multiples of $360^{\circ}$ (in degrees) or $2\pi$ (in radians).

Step2: Find positive coterminal angles in degrees for $\theta = 225^{\circ}$

Add $360^{\circ}$ once: $225^{\circ}+360^{\circ}=585^{\circ}$
Add $360^{\circ}$ twice: $225^{\circ}+ 2\times360^{\circ}=225^{\circ}+720^{\circ}=945^{\circ}$

Step3: Find negative coterminal angles in degrees for $\theta = 225^{\circ}$

Subtract $360^{\circ}$ once: $225^{\circ}-360^{\circ}=-135^{\circ}$
Subtract $360^{\circ}$ twice: $225^{\circ}-2\times360^{\circ}=225^{\circ}-720^{\circ}=-495^{\circ}$

Step4: Find positive coterminal angles in radians for $\theta=-\frac{5\pi}{4}$

Add $2\pi$ once: $-\frac{5\pi}{4}+2\pi=-\frac{5\pi}{4}+\frac{8\pi}{4}=\frac{3\pi}{4}$
Add $2\pi$ twice: $-\frac{5\pi}{4}+2\times2\pi=-\frac{5\pi}{4} + 4\pi=-\frac{5\pi}{4}+\frac{16\pi}{4}=\frac{11\pi}{4}$

Step5: Find negative coterminal angles in radians for $\theta =-\frac{5\pi}{4}$

Subtract $2\pi$ once: $-\frac{5\pi}{4}-2\pi=-\frac{5\pi}{4}-\frac{8\pi}{4}=-\frac{13\pi}{4}$
Subtract $2\pi$ twice: $-\frac{5\pi}{4}-2\times2\pi=-\frac{5\pi}{4}-4\pi=-\frac{5\pi}{4}-\frac{16\pi}{4}=-\frac{21\pi}{4}$

Answer:

For $225^{\circ}$: Positive - $585^{\circ},945^{\circ}$; Negative - $-135^{\circ},-495^{\circ}$
For $-\frac{5\pi}{4}$: Positive - $\frac{3\pi}{4},\frac{11\pi}{4}$; Negative - $-\frac{13\pi}{4},-\frac{21\pi}{4}$