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
    • Quadrant IV (270°–360°): \sin
  • 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{vy}{vx}\right) using \tan definition.
  • Kinematics & Projectile Motion:
    • Breaking an initial velocity into vx=v\cos\theta and vy=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.