Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$3 \cos ^ { 2 } \theta + 2 \cos \theta = 1$

Question

Quizplus · Accepted Answer

Explanation:Factor the left side of the equation:$$3 \cos ^ { 2 } 	heta + 2 \cos 	heta = 1$$$$( 3 \cos 	heta + 1 ) ( \cos 	heta - 1 ) = 0$$Set each factor equal to 0 and solve for θ:$$3 \cos 	heta + 1 = 0$$$$\cos 	heta = - \frac { 1 } { 3 }$$$$	heta = \cos ^ { - 1 } \left( - \frac { 1 } { 3 } ight) \approx 103.2 ^ { \circ }$$$$	heta = 103.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$$$\cos 	heta - 1 = 0$$$$\cos 	heta = 1$$$$	heta = \cos ^ { - 1 } ( 1 ) = 0 ^ { \circ }$$$$	heta = 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$Therefore, the solutions are:$$	heta = 103.2 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$$$	heta = 0 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$Converting to radians:$$	heta = 103.2 ^ { \circ } \left( \frac { \pi } { 180 ^ { \circ } } ight) \approx 1.800 \mathrm { radians }$$$$	heta = 0 ^ { \circ } \left( \frac { \pi } { 180 ^ { \circ } } ight) = 0 \mathrm { radians }$$Therefore, the solutions are:$$	heta = 1.800 \mathrm { radians } + 2 \pi \mathrm { n }$$$$	heta = 0 \mathrm { radians } + 2 \pi \mathrm { n }$$$$	heta = 70.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$$$	heta = 180 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$$$	heta = 289.5 ^ { \circ } + 360 ^ { \circ } \mathrm { n }$$

Solve the Equation (X in Radians and θ in Degrees) 3cos⁡2θ+2cos⁡θ=13 \cos ^ { 2 } \theta + 2 \cos \theta = 13cos2θ+2cosθ=1

Solve the Equation (X in Radians and θ in Degrees) $3 \cos ^ { 2 } \theta + 2 \cos \theta = 1$