Squares Erase the Signs
Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 2 of 52.
Past 90° the point leaves the first quadrant and its coordinates can turn negative. At 150° the point is (-√(3)/2, 1/2): left of the y-axis, so its x is negative. The triangle under it still has legs of length √(3)/2 and 1/2, and squaring a negative number makes it positive: (-√(3)/2)² = 3/4. So the squares still add to 1, in every quadrant.
x is negative here, but x² is not.
That is why the identity can find one coordinate from the other, but not its sign. Say sin(x) = 3/5 and x is in quadrant II. Then cos(x)² = 1 - 9/25 = 16/25, so cos x is 4/5 or -4/5. Quadrant II is left of the y-axis, so cos(x) = -4/5. The identity gives the size; the quadrant gives the sign.
Challenge
The angle 225° ends in quadrant III. Set the coordinates of its point on the circle, signs included. Then press Check.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Trig Identities
1Identities on the Circle
sin²x + cos²x = 1Squares Erase the SignsIdentity or Equation?Recap: the Circle Identity4Even, Odd and Cofunctions
Negative AnglesEven and Odd in a ProofEven and Odd GraphsCofunctionsRecap: Signs of Angles2Reciprocal and Quotient
The Quotient IdentitySecant, Cosecant, CotangentDividing by a FractionWhere They BreakRecap: Reciprocals5Proving Identities
Work on One SideRewrite in sin and cosFactor FirstMake One FractionProofs That Go WrongRecap: Prove It8Half Angles, Power Reduction
Power Reductionsin²x Is a WaveHalf-Angle Valuestan of a Half AngleRecap: Half Angles