5.2 Trigonometric Ratios (SOH-CAH-TOA)

Where the Ratios Come From

Standard G-SRT.6 builds trigonometry from similarity: because all right triangles that share an acute angle are similar (by AA), their corresponding side ratios are equal. That means each ratio depends only on the angle, not on how big the triangle is. Those fixed ratios are the trigonometric functions, and they are named for the sides they compare.

Defining the Three Ratios: SOH-CAH-TOA

Relative to a chosen acute angle (call it θ), the three sides are the opposite leg, the adjacent leg, and the hypotenuse. The mnemonic SOH-CAH-TOA encodes all three primary ratios.

The hypotenuse never changes, but "opposite" and "adjacent" swap depending on which acute angle you pick. Label the triangle before choosing a ratio: mark the right angle, mark your reference angle, then tag the remaining two sides as opposite and adjacent.

Finding a Missing Side

When you know one acute angle and one side, pick the ratio that connects the side you have to the side you want.

Example: unknown in the numerator

A right triangle has a 40° angle and a hypotenuse of 10. Find the side opposite the 40° angle. Opposite and hypotenuse point to sine: sin 40° = opp / 10, so opp = 10 · sin 40° ≈ 10(0.6428) ≈ 6.43. To find the adjacent side instead, use cosine: adj = 10 · cos 40° ≈ 7.66.

Example: unknown in the denominator

Suppose the leg opposite a 35° angle is 12 and you need the adjacent leg x. Then tan 35° = 12 / x. When the unknown is on the bottom, multiply both sides by x and divide by the ratio: x = 12 / tan 35° ≈ 12 / 0.7002 ≈ 17.14.

Finding a Missing Angle

If you know two sides, you can recover the angle with an inverse trigonometric function (sin⁻¹, cos⁻¹, tan⁻¹, also written arcsin, arccos, arctan). Set up the ordinary ratio, then apply the inverse.

Worked example

Angle A has an opposite leg of 9 and an adjacent leg of 12. Opposite over adjacent is tangent, so tan A = 9/12 = 0.75, and A = tan⁻¹(0.75) ≈ 36.87°. If instead you were given the opposite leg and the hypotenuse, you would use sin⁻¹; adjacent and hypotenuse call for cos⁻¹.

Same Triangle, Two Reference Angles

Because opposite and adjacent are defined by the angle you choose, one triangle produces two different labelings. Consider a right triangle with legs 3 and 4 and hypotenuse 5. From the angle whose opposite leg is 3, sin = 3/5, cos = 4/5, and tan = 3/4. From the other acute angle, the roles flip: its opposite leg is 4, so sin = 4/5, cos = 3/5, and tan = 4/3. Notice that the sine of one angle equals the cosine of the other, a preview of the cofunction rule in Section 5.4.

Finding the Hypotenuse

The hypotenuse can also be the unknown. If a 28° angle has an opposite leg of 15, then sin 28° = 15 / hyp, so hyp = 15 / sin 28° ≈ 15 / 0.4695 ≈ 31.9. Whenever the side you know and the side you want straddle the ratio as numerator and denominator, isolate the unknown algebraically by multiplying or dividing rather than guessing which operation to use.

Exact Values Worth Memorizing

The special triangles from Section 5.1 give exact ratios that sometimes appear as answer choices.

If a question states cos θ = 1/2 and asks for θ, recognizing the table gives θ = 60° with no calculator work. Likewise, tan θ = 1 signals the 45° angle, and sin θ = √3/2 signals 60°. Reading these backward from a ratio to an angle is a fast, error-free check whenever the answer choices are clean special-angle values.

Calculator: Degree Mode Is Mandatory

The Regents reports angles in degrees, so your graphing calculator must be in DEGREE mode, not radians. A single wrong-mode setting turns every trig answer into nonsense, so check the mode at the start of the exam.

• sin 30° = 0.5 in degree mode; if your calculator returns about −0.988, it is in radians.
• Use the calculator's inverse keys (often 2nd then sin/cos/tan) for angles.
• Round only at the final step, and match the precision the question asks for (nearest degree, nearest tenth, and so on).

Common Traps

• Mislabeling opposite and adjacent: they are defined relative to the reference angle, so re-tag them whenever the angle changes.
• Reaching for a plain ratio when the unknown is an angle; angles require the inverse function.
• Leaving the calculator in radian mode.
• Rounding intermediate values, which snowballs into a wrong final digit; keep full precision until the end.