Swap cos²x for sin²x
Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 49 of 56.
2cos(x)² + sin(x) = 1 has two functions, cosine squared and sine. sin(x)² + cos(x)² = 1 lets you trade one for the other: cos²x = 1 − sin²x.
Substitute and expand: 2 − 2sin(x)² + sin(x) = 1, so 2sin(x)² − sin(x) − 1 = 0, a quadratic in sin x: (2sin(x) + 1)(sin(x) − 1) = 0. sin x = −1/2 gives 210° and 330°; sin x = 1 gives 90°.
Challenge
Solve 2sin(x)² + 3cos(x) + 3 = 0 for every x in [0°, 360°), then press Check.
Try it yourself
2sin(x)^2 + 3cos(x) + 3 = 0
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: Compositions7General Solutions
Every SolutionTangent's Single FamilyFamilies in RadiansFamilies, RoundedRecap: Every Solution10Identities First
Write sin 2x OutPick the Right cos 2xSwap cos²x for sin²xsec, csc and cotRecap: Identities First2Inverses 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 Domain