Revision sheet

Radians

• $180^\circ = \pi$ rad.
• Degrees → radians: $\times \dfrac{\pi}{180}$. Radians → degrees: $\times \dfrac{180}{\pi}$.
• Arc length: $s = r\theta$, $\theta$ in radians.

Watch out: never put degrees into $s = r\theta$.

The unit circle

• The point at angle $\theta$ is $(\cos\theta, \sin\theta)$.
• $\dfrac{\pi}{6}$: $\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$; $\dfrac{\pi}{4}$: $\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$; $\dfrac{\pi}{3}$: $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$.

Watch out: cosine is $x$, sine is $y$.

Reference angles

1. Find the quadrant.
2. Reference angle: acute angle to the $x$-axis ($180^\circ - \theta$, $\theta - 180^\circ$ or $360^\circ - \theta$).
3. Value of the reference angle, with the quadrant's sign.

Watch out: cosine is negative on the left (II, III), sine below (III, IV).

Tangent and reciprocals

• $\tan = \dfrac{\sin}{\cos}$, $\cot = \dfrac{\cos}{\sin}$.
• $\sec = \dfrac{1}{\cos}$, $\csc = \dfrac{1}{\sin}$.

Watch out: sec goes with cos, csc with sin.

Right-triangle trigonometry

• SOH CAH TOA.
• Side from an angle: multiply ($\text{opposite} = \text{hyp} \cdot \sin\theta$).
• Angle from two sides: $\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$.

Watch out: name the sides from the angle you use.

Area of any triangle

½ab sin C, with C between a and b.

Watch out: use the included angle.

Law of sines

a / sin A = b / sin B = c / sin C.

Watch out: pair each side with its opposite angle.

Law of cosines

c² = a² + b² − 2ab cos C.

Watch out: Pythagoras alone works only for a right angle.

Sinusoidal graphs $y = A\sin(Bx) + D$

• Amplitude $|A|$, midline $y = D$.
• Period $\dfrac{2\pi}{B}$.
• Maximum $D + |A|$, minimum $D - |A|$.

Watch out: a bigger $B$ means a shorter period.

Phase shift and models

• $A\sin\big(B(x - h)\big) + D$: shift $h$ right.
• $\sin(Bx - C)$: shift $\dfrac{C}{B}$, after factoring out $B$.
• Model: $A$ = half the range, $D$ = middle value, $B = \dfrac{2\pi}{\text{period}}$.

Watch out: factor out $B$ before reading the shift.

Inverse trig functions

• $\sin^{-1}$: angle in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$. $\tan^{-1}$: in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.
• $\cos^{-1}$: angle in $[0, \pi]$.

Watch out: $\sin^{-1}x$ is an angle, not $\dfrac{1}{\sin x}$.