Trigonometry

A ladder leans against a wall. The ladder is 10 feet long and makes a 60° angle with the ground. How high up the wall does it reach?

You don’t need a tape measure — you need trigonometry. The ladder is the hypotenuse, the wall height is the opposite side, and sine connects them: height = $10\sin 60° = 10 \times 0.87 \approx 8.7$ feet. Three seconds, no calculator — because you memorized that $\sin 60° \approx 0.87$.

On the MCAT, trigonometry appears in vector decomposition, inclined plane problems, optics (Snell’s law), and any problem involving angles. The test almost exclusively uses five special angles, so mastering trig is really just memorizing a small table.

SOH CAH TOA

For a right triangle with angle θ:

The Five Special Angles

This is the only trig table you need on the MCAT. Memorize it cold.

Reading the Table Efficiently

Notice the sine values follow a clean pattern using square roots:

• sin 0 = sqrt(0)/2 = 0
• sin 30 = sqrt(1)/2 = $\frac{1}{2}$
• sin 45 = sqrt(2)/2
• sin 60 = sqrt(3)/2
• sin 90 = sqrt(4)/2 = 1

The numerator under the square root just counts: 0, 1, 2, 3, 4. The denominator is always 2. Cosine is the same sequence in reverse order.

The Unit Circle

The unit circle extends trig beyond 0-90 degrees to all angles. It is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the coordinates of the point on the circle are (cos θ, sin θ).

Key quadrant signs:

• Quadrant I (0-90 degrees): sin positive, cos positive
• Quadrant II (90-180 degrees): sin positive, cos negative
• Quadrant III (180-270 degrees): sin negative, cos negative
• Quadrant IV (270-360 degrees): sin negative, cos positive

Inverse Trig Functions

Inverse trig functions answer: “What angle gives this ratio?”

• If sin θ = 0.5, then θ = arcsin(0.5) = 30 degrees
• If cos θ = 0.71, then θ = arccos(0.71) = 45 degrees
• If tan θ = 1, then θ = arctan(1) = 45 degrees

On the MCAT, you will almost always be able to recognize the angle from your memorized table rather than needing to compute an inverse trig function from scratch.

Common MCAT Applications

Vector components: Fx = F cos θ, Fy = F sin θ (where θ is measured from the x-axis)

Inclined planes: Weight components on a ramp at angle θ: parallel = mg sin θ, perpendicular = mg cos θ

Snell’s Law: n1 sin θ_1 = n2 sin θ_2

Projectile motion: $v_{x}$ = v cos θ, $v_{y}$ = v sin θ