Prove symmetry by moving a unit-circle point

Try

Mark the point P=(3/5,4/5) on a circle of radius one centered at the origin. Its coordinates satisfy (3/5)²+(4/5)²=1. A half-turn moves it to (−3/5,−4/5); reflection across the horizontal axis moves it to (3/5,−4/5). You can find both points without knowing the original angle as a decimal.

At angle x, measured in radians from the positive horizontal axis, the circle point is (cos x,sin x). Cosine means horizontal coordinate and sine means vertical coordinate. One full turn is 2π radians, so the transformations above connect general angle expressions to coordinate changes.

Predict

Predict the sine and cosine at π+x and 2π−x. Which transformation changes both signs? Which preserves the horizontal coordinate? Before calculating tangent, check whether the horizontal coordinate is zero. Tangent is a ratio, so a point on the vertical axis needs separate treatment.

Experiment

The diagram uses the same scale on both axes and shows the representative point c=3/5, s=4/5. Its generic labels display the coordinate rule: a half-turn sends P=(c,s) to H=(−c,−s), while horizontal-axis reflection sends it to R=(c,−s). Thus the pictured points are P=(3/5,4/5), H=(−3/5,−4/5) and R=(3/5,−4/5). Recreate the construction for P=(−12/13,5/13), first predicting quadrant signs. Then complete this general table rather than drawing only first-quadrant examples.

Angle	Point if the original point is (c,s)	Tangent when c≠0
x	(c,s)	s/c
π+x	(−c,−s)	s/c
2π−x	(c,−s)	−s/c

Check the special starting point (0,1). Its transformed points still exist, but the tangent entries remain undefined. Never turn “undefined on both sides” into an equality of numerical values.

Understand

A half-turn negates both coordinates for any unit-circle point. Thus cos(π+x)=−cos x and sin(π+x)=−sin x for every real x. When cos x≠0, their quotient gives tan(π+x)=tan x. This proof does not require x to lie between zero and π/2; a signed or multi-turn angle ends at a point to which the same transformation applies.

Reflection across the horizontal axis reverses the direction of angular travel: the point for −x is (cos x,−sin x). Therefore cosine is even and sine is odd. An even function satisfies f(−x)=f(x); an odd function satisfies f(−x)=−f(x), on a domain symmetric about zero. Tangent is odd on its symmetric domain, which excludes π/2+kπ for every integer k. Adding a full turn to −x gives 2π−x without changing the endpoint, explaining the third row of the table.

A period P is a positive shift that preserves both the domain and every function value. Full turns prove that 2π is a period of sine and cosine. Half-turns preserve the tangent ratio and its domain, proving that π is a tangent period. Repeating the argument gives integer multiples. Equal function values at two isolated angles do not prove a period: a period must work at every admissible angle.

To justify the least positive periods, cosine repeating at zero requires cos P=1, whose circle point is (1,0), so positive candidates are whole turns. For sine, repeating at zero first restricts P to integer multiples of π; repeating at π/2 rejects odd multiples. For tangent, repeating at zero requires tan P=0, whose admissible points lie on the horizontal axis, so positive candidates are multiples of π. These arguments establish minimality as well as an identity.

Apply

Repair “−π/4 equals 7π/4.” They differ by 2π as real numbers but have the same circle endpoint. Their cosine and sine agree. Explain why this does not make either of them the same angle value returned by every principal inverse function.

Continue

Use the same coordinate approach to derive angle-addition formulas next. If radians or special triangles need review, revisit radians and the unit circle. A general transformation proof supports infinitely many angles; checking several special values only tests examples.