Trig Values of Special Angles

Calculus Readiness · Axiom Academy

The exact sine, cosine, and tangent at 30°, 45°, and 60° — read straight off two reference triangles sitting inside the unit circle. Start with an isosceles right triangle : two equal legs and a right angle between them. Call each leg 1 . By the Pythagorean theorem the hypotenuse is . Now shrink that triangle until its hypotenuse is exactly the radius of the unit circle, tilted to . The tip of the triangle lands on the circle, and its coordinates are precisely . The point at 45° on the unit circle Take an equilateral triangle with side 2 — every angle — and drop a line straight down the middle. That splits it into two congruent right triangles, each with angles . The cut halves the base, so the short leg (opposite ) is 1 , the hypotenuse stays 2 , and the long leg (opposite ) is . Scale the hypotenuse to the unit radius and the same triangle reads off both and on the circle. Opposite the angle — half of the original side 2 . Opposite the angle — found from . The original equilateral side; it becomes the radius when scaled to 1 . Read from the steep tip, from the shallow tip — same sides, swapped roles. Tangent is opposite over adjacent. At : . At the roles flip: . 3. The Whole Quarter-Turn at Once Put the two triangles together and sweep one hand from to . At each special angle the point on the circle has coordinates — the cosine is how far right , the sine how far up . Watch the five exact pairs stamp themselves onto the circle in order; that is the master table, drawn.

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