Trigonify

Topics · 4 identities

Basics

The core ratios, degree-radian conversion, and circle measures every trig course opens with.

SOHCAHTOA: the sine, cosine and tangent ratios

sin⁡θ=opphypcos⁡θ=adjhyptan⁡θ=oppadj\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\quad \cos\theta = \dfrac{\text{adj}}{\text{hyp}}\quad \tan\theta = \dfrac{\text{opp}}{\text{adj}}

SOHCAHTOA is the mnemonic for the three ratios in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Everything else in trigonometry is built on these three. Use it whenever a problem gives you a right angle and any two of a side, an angle, or the hypotenuse.

Open identity

Degrees to radians: 1° = π/180 radians

1∘=π180 rad1 rad=180π ⁣ ⁣∘1^{\circ} = \dfrac{\pi}{180}\text{ rad}\qquad 1\text{ rad} = \dfrac{180}{\pi}^{\!\!\circ}

One degree is π/180 radians, and one radian is 180/π degrees - so multiply by π/180 going into radians and by 180/π coming out. Radians are the unit calculus assumes, which is why arc length, sector area and every derivative of a trig function require them. Getting caught in degree mode is the single most common source of wrong answers on a calculator.

Open identity

Arc length formula: s = rθ

s=rθs = r\theta

The arc cut off by a central angle θ on a circle of radius r has length s = rθ. θ must be in radians - the formula is only this clean because a radian is defined as the angle whose arc equals the radius. In degrees you would need s = (πrθ)/180 instead.

Open identity

Sector area formula: A = ½r²θ

A=12r2θA = \tfrac{1}{2} r^{2} \theta

A sector with central angle θ on a circle of radius r has area ½r²θ, with θ in radians. It is the whole circle's πr² scaled by the fraction θ/2π that the sector occupies. Pair it with the arc-length formula for any "slice of a circle" question.

Open identity