Trigonometry Essentials for the MCAT

Definitions & Core Relationships

• Right triangles are the ONLY triangles tested on the MCAT for trigonometry questions.

  • Focus is almost always on the two “special” right triangles: $30^{\circ} ext{-}60^{\circ} ext{-}90^{\circ}$ and $45^{\circ} ext{-}45^{\circ} ext{-}90^{\circ}$.

• Naming the triangle in the transcript:

  • Legs: $a$ (opposite the angle of interest) and $b$ (adjacent to the angle of interest)
  • Hypotenuse: $c$

• Primary trigonometric ratios (SOH-CAH-TOA mnemonic):

  • $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}=\dfrac{a}{c}$
  • $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}=\dfrac{b}{c}$
  • $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac{a}{b}$

• Domains (possible output values):

  • $-1\le\sin\theta\le1$
  • $-1\le\cos\theta\le1$
  • $\tan\theta\in(-\infty,\infty)$ (can take any real value)

Inverse (Arc) Functions

• Purpose: convert a trigonometric ratio back into an angle.
  • $\sin^{-1}(x)=\arcsin(x)\Rightarrow\theta$
  • $\cos^{-1}(x)=\arccos(x)\Rightarrow\theta$
  • $\tan^{-1}(x)=\arctan(x)\Rightarrow\theta$
• MCAT–relevant usage:
  • Determining the direction (angle) of a resultant vector in physics (vector addition/subtraction problems).
  • Example in the transcript’s figure (generic): $\sin^{-1}\left(\dfrac{a}{c}\right)=\theta$.

Special Right Triangles & Must-Know Values

• You MUST either memorize or be able to re-derive these quickly on test day.

• 30-60-90 triangle (side ratios $1:\sqrt{3}:2$):

  • $\sin30^{\circ}=\dfrac12$
  • $\cos30^{\circ}=\dfrac{\sqrt3}{2}$
  • $\tan30^{\circ}=\dfrac{\sqrt3}{3}$
  • $\sin60^{\circ}=\dfrac{\sqrt3}{2}$
  • $\cos60^{\circ}=\dfrac12$
  • $\tan60^{\circ}=\sqrt3$

• 45-45-90 triangle (side ratios $1:1:\sqrt2$):

  • $\sin45^{\circ}=\dfrac{\sqrt2}{2}$
  • $\cos45^{\circ}=\dfrac{\sqrt2}{2}$
  • $\tan45^{\circ}=1$

• Supplemental angles frequently tested in circular motion & waves:

  • $\theta=0^{\circ}$: $\sin0^{\circ}=0$, $\cos0^{\circ}=1$, $\tan0^{\circ}=0$
  • $\theta=90^{\circ}$: $\sin90^{\circ}=1$, $\cos90^{\circ}=0$, $\tan90^{\circ}$ is undefined (division by zero – vertical asymptote)
  • $\theta=180^{\circ}$: $\sin180^{\circ}=0$, $\cos180^{\circ}=-1$, $\tan180^{\circ}=0$

Ranges, Signs & Quadrants (implicit in transcript)

• While not explicitly discussed, the sign conventions matter:
  • Quadrant I (0°–90°): $\sin,\cos,\tan>0$.
  • Quadrant II (90°–180°): $\sin>0,\cos<0,\tan<0$.
  • Quadrant III (180°–270°): $\sin<0,\cos<0,\tan>0$.
  • Quadrant IV (270°–360°): $\sin<0,\cos>0,\tan<0$.
• Ranges corroborate the statement that $\sin,\cos\in[-1,1]$ and $\tan$ spans all real numbers except where undefined.

Practical MCAT Connections

• Physics Vectors:
  • Resultant vector angle: $\theta=\tan^{-1}\left(\dfrac{v<em>y}{v</em>x}\right)$ using $\tan$ definition.
• Kinematics & Projectile Motion:
  • Breaking an initial velocity into $v<em>x=v\cos\theta$ and $v</em>y=v\sin\theta$ uses memorized trig values for quick mental math.
• Simple Harmonic Motion & Waves:
  • Phase relationships (e.g., displacement vs. acceleration) rely on $\sin$ and $\cos$ values at multiples of $\pi/2$ ($90^{\circ}$).

Key Takeaways & Test-Day Strategy

• Memorize the two special right triangles OR practice drawing them from scratch in <10 seconds.
• Know the output ranges so you can spot impossible answer choices quickly (e.g., $\sin\theta=1.3$ is impossible).
• Remember that $\tan$ becomes undefined when $\cos\theta=0$ (odd multiples of $90^{\circ}$).
• When asked for an angle, expect to employ inverse functions, especially $\arctan$ for vector problems.