A Quadratic in Sine
Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 43 of 56.
Solve 2cos(x)² − 3cos(x) + 1 = 0. With u for cos x it is 2u² − 3u + 1 = 0, which factors as (2u − 1)(u − 1): multiply back to check, 2u² − 2u − u + 1.
Put the cosine back and split: 2cos x − 1 = 0 gives cos x = 1/2, so 60° or 300°, and cos x − 1 = 0 gives cos x = 1, so 0° only, since 360° is outside the domain. Three answers.
Challenge
Solve 2sin(x)² + sin(x) − 1 = 0 for every x in [0°, 360°). Factor it as a quadratic in the sine, then solve each case, and press Check.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Equations and Inverses
1Your First Trig Equations
One Equation, Two AnglesReading Signs on the CircleThe Reference AngleRecap: First Equations4Compositions
Draw the TriangleThe Outer FunctionA Negative InsideWhen Undoing Does Not UndoRecap: Compositions2Inverses and Their Ranges
Why arcsin Gives One AngleNegative Inputs to arcsinThe Range of arccosThe Range of arctanRecap: Inverse Ranges5Equations on a Domain
A Negative CosineTangent EquationsEquations in RadiansTwo Turns, Four AnswersRecap: On a Domain3Evaluating Inverse Trig
Rounded Angles in DegreesRounded Angles in Radiansarctan, RoundedAn Angle From a SlopeRecap: Inverse Values6Off the Special Angles
Both Answers, RoundedA Negative Sine, RoundedCosine, RoundedTangent, RoundedRounded, in RadiansRecap: Off the Table9Factoring Trig Equations
A Common FactorNever Divide by sin xA Quadratic in SineA Case With No AngleSquare Roots Give Two SignsRecap: Factoring