Trigonometry

Trigonometry on the MCAT centers on right triangles: using SOH CAH TOA to relate an angle to the ratios of a triangle's sides, knowing the standard sine/cosine/tangent values at 0°, 30°, 45°, 60°, and 90°, and recognizing the two special right triangles — 30-60-90 and 45-45-90 — that recur throughout physics problems involving vectors and forces.

Key Takeaways

• In a right triangle relative to angle θ: hypotenuse (longest side, opposite the 90° angle), opposite (across from θ), and adjacent (next to θ, not the hypotenuse).

• SOH CAH TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

• Inverse trig functions aren't calculator-free on the MCAT, but appear conceptually in vector-direction problems.

• Memorize sine, cosine, and tangent at 0°, 30°, 45°, 60°, and 90° — they recur constantly.

• The 30-60-90 triangle (from an equilateral triangle) has side ratio 1 : √3 : 2; the 45-45-90 triangle (from a square) has side ratio 1 : 1 : √2.

Right Triangle Basics

Trigonometry studies the relationships between the angles and sides of triangles — specifically right triangles, which always contain one 90° angle.

Relative to a chosen angle θ (theta), the three sides of a right triangle have specific names:

• Hypotenuse — the longest side, directly opposite the 90° angle.

• Opposite side — the side opposite angle θ.

• Adjacent side — the side next to angle θ, excluding the hypotenuse.

SOH CAH TOA

The three primary trigonometric functions — sine, cosine, and tangent — define fixed ratios between these sides. The mnemonic SOH CAH TOA captures all three:

• SOH: sin θ = opposite/hypotenuse

• CAH: cos θ = adjacent/hypotenuse

• TOA: tan θ = opposite/adjacent

Inverse Trig Functions

Each of these three functions has an inverse (arcsine, arccosine, arctangent), which recovers the angle from a ratio of sides. You won't be expected to calculate inverse trig functions without a calculator on the MCAT — but they're likely to show up conceptually in questions asking for the direction of a resultant vector after vector addition or subtraction.

Standard Trig Values

The sine, cosine, and tangent values at five specific angles — 0°, 30°, 45°, 60°, and 90° — come up often enough to be worth knowing directly:

Standard angle values

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

The 30-60-90 Triangle

The 30-60-90 triangle is a special right triangle formed by splitting an equilateral triangle (all angles 60°, all sides equal) down the middle. This produces a right triangle with:

• One 30° angle,

• One 60° angle (unchanged from the original),

• A hypotenuse equal to the original equilateral triangle's side length.

Its sides always follow a fixed ratio of 1 : √3 : 2:

• The shortest side, opposite the 30° angle, has length 1.

• The longer leg, opposite the 60° angle, has length √3.

• The hypotenuse, the original equilateral triangle's side, has length 2.

The 45-45-90 Triangle

The 45-45-90 triangle is formed by cutting a square diagonally in half, producing a right triangle with two equal 45° angles. Its sides always follow a fixed ratio of 1 : 1 : √2:

• The two legs are equal in length (1), since they come from the square's equal sides.

• The hypotenuse — the diagonal of the square — has length √2.

Common MCAT Mistakes

• Mixing up opposite and adjacent. Both are relative to the chosen angle θ, not fixed sides of the triangle — the same side can be "opposite" for one angle and "adjacent" for the other.

• Forgetting tan 90° is undefined, not zero. Since tan θ = opposite/adjacent and the adjacent side shrinks to zero at 90°, the ratio blows up rather than vanishing.

• Mislabeling the 30-60-90 side ratio. The sides scale as 1 : √3 : 2 in a fixed order — shortest side (opposite 30°), longer leg (opposite 60°), hypotenuse — not 1 : 2 : √3.

• Assuming the 45-45-90 hypotenuse equals the leg length. The hypotenuse is √2 times a leg, not equal to it — a direct consequence of the two legs being equal.

MCAT-Style Concept Check

Question: A right triangle has a 60° angle and a hypotenuse of length 10. Using the 30-60-90 side ratio, what is the length of the side opposite the 60° angle?

• A) 5

• B) 5√2

• C) 5√3

• D) 10√3

Answer: C

Explanation: In a 30-60-90 triangle, the sides scale as 1 : √3 : 2, with the hypotenuse corresponding to 2. Here the hypotenuse (10) corresponds to 2, so the scale factor is 5. The side opposite the 60° angle corresponds to √3 in the ratio, giving 5√3. Answer A gives the side opposite the 30° angle instead, answer B uses the 45-45-90 ratio, and answer D doubles the correct scale factor.

FAQ

What does SOH CAH TOA stand for?

SOH CAH TOA is a mnemonic for the three primary trig ratios: sin θ = opposite/hypotenuse (SOH), cos θ = adjacent/hypotenuse (CAH), and tan θ = opposite/adjacent (TOA).

Do I need to calculate inverse trig functions on the MCAT?

No — the MCAT is calculator-free, so you won't be asked to compute arcsine, arccosine, or arctangent directly. They do show up conceptually, though, especially in questions about the direction of a resultant vector.

Which sine, cosine, and tangent values should I have memorized?

The values at 0°, 30°, 45°, 60°, and 90° come up repeatedly enough to be worth memorizing directly, rather than re-deriving them each time.

How are the 30-60-90 and 45-45-90 triangles formed?

The 30-60-90 triangle comes from splitting an equilateral triangle in half, giving a fixed side ratio of 1 : √3 : 2. The 45-45-90 triangle comes from cutting a square diagonally in half, giving a fixed side ratio of 1 : 1 : √2.