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Algebra readiness

Use radians and triangle ratios · Diagrams

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Circular measure and right-triangle ratios

Illustration: r = AC = 3 units.

Acute angle θ45° = π/4 rad
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Arc length

s=rθs=r\theta

θ in radians
OPQrθs

The highlighted arc PQ subtends θ at centre O.

s ≈ 2.3562 units.

Sector area

A=12r2θA=\tfrac12r^2\theta

θ in radians
OPQrθ

The shaded region is sector POQ, with area A.

A ≈ 3.5343 square units.

Right-triangle ratios

The right angle is at B; the acute angle θ is at C. AB is opposite, BC adjacent and AC the hypotenuse.

ABCABBCACθ

AB ≈ 2.1213, BC ≈ 2.1213, AC = 3 units.

sin⁡θ=ABAC\sin\theta=\dfrac{AB}{AC}

≈ 0.7071

cos⁡θ=BCAC\cos\theta=\dfrac{BC}{AC}

≈ 0.7071

tan⁡θ=ABBC\tan\theta=\dfrac{AB}{BC}

≈ 1

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Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
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