2.2 Trigonometry Basics

Right-triangle trigonometry shows up whenever the NAPT presents an angle instead of a straight length — an incline, a sightline, or a support brace set at a known angle. This section covers the three basic trig ratios, the SOH-CAH-TOA mnemonic, the two special right triangles worth memorizing, angle-of-elevation and angle-of-depression problems, and how to solve for a missing side or angle.

Sides of a Right Triangle Relative to an Angle

Every right triangle has one 90° angle. Once you pick one of the other two acute angles as your reference angle (labeled θ, the Greek letter theta), the three sides get renamed relative to that angle:

• The hypotenuse is always the longest side, opposite the right angle — this label never changes.
• The opposite side is the leg directly across from θ.
• The adjacent side is the leg that touches θ (other than the hypotenuse).

SOH-CAH-TOA

Trigonometry is the study of the relationships between a triangle's angles and its side lengths. For a right triangle, three ratios connect θ to the side lengths, remembered with the mnemonic SOH-CAH-TOA:

Special Right Triangles

Two triangle shapes appear often enough to memorize outright, saving time on test day:

• 45-45-90 triangle: the two legs are equal, and the hypotenuse equals a leg times √2 (√2 ≈ 1.414). If each leg is 5, the hypotenuse is 5 × 1.414 ≈ 7.07.
• 30-60-90 triangle: the sides are always in the ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). If the short leg is 4, the long leg is 4 × √3 ≈ 6.93 and the hypotenuse is 8.

Solving for a Missing Side

If you know one acute angle and one side, you can find either remaining side by choosing the ratio that connects the side you know to the side you want.

Worked Example 1

A loading ramp forms a 30° angle with the ground. The ramp itself (the hypotenuse) is 20 feet long. How high off the ground does the top of the ramp rise?

The height is the side opposite the 30° angle, and we know the hypotenuse — that calls for the sine ratio (SOH):

sin 30° = opposite / hypotenuse 0.5 = opposite / 20 opposite = 0.5 × 20 = 10 feet

Angle of Elevation and Angle of Depression

Many NAPT trigonometry problems are dressed up as sighting problems. The angle of elevation is measured upward from horizontal to an object above the observer — a lookout spotting the top of a mast. The angle of depression is measured downward from horizontal to an object below the observer — a sailor watching a boat approach the hull. Both are measured from the horizontal, never from the ground or vertical — that mix-up is a common trap, and a lower observer's elevation angle equals the higher point's depression angle back down, since the two horizontal reference lines are parallel.

Worked Example 2

A sailor on a pier measures a 35° angle of elevation to the top of a nearby mast, 60 feet away horizontally, at the mast's base height. How tall is the mast, to the nearest foot?

The 60 feet is adjacent to the angle and the mast height is opposite it, so use tangent (TOA):

tan 35° = opposite / adjacent 0.700 ≈ opposite / 60 opposite = 60 × 0.700 = 42 feet

Tangent applies here, not sine, because neither known length is the hypotenuse.

Solving for a Missing Angle

If you know two sides but not the angle, use the inverse trig function — arcsin, arccos, or arctan — which undoes the ratio and returns the angle itself.

Worked Example 3

A right triangle has an opposite side of 8 feet and an adjacent side of 15 feet, relative to angle θ. What is θ?

Since we know the opposite and adjacent sides, use tangent (TOA):

tan θ = opposite / adjacent = 8 / 15 ≈ 0.533

θ = arctan(0.533) ≈ 28°

A calculator or trig table is needed for non-special angles like this one — Chapter 8 covers calculator strategy for test day.

Checking Your Work with the Pythagorean Theorem

Once trig gives one missing side, the Pythagorean theorem (a² + b² = c²) offers an independent check. In the mast example, adjacent = 60 feet and opposite = 42 feet, so the hypotenuse — the sailor's line-of-sight distance to the masthead — is:

c = √(60² + 42²) = √5,364 ≈ 73.2 feet

That matches 60 / cos 35° ≈ 73.3 feet within rounding. Expect NAPT questions that combine a trig ratio with the Pythagorean theorem or an area formula in one problem.

Common Angle Values

For the special angles, sine, cosine, and tangent have fixed values worth recognizing on sight:

Avoiding the Most Common Trig Error

The single most common mistake on right-triangle trig problems is mislabeling opposite and adjacent relative to the marked angle — the same leg can be opposite for one acute angle and adjacent for the other acute angle in the very same triangle. Before writing any ratio, re-read the problem, confirm exactly which angle is θ, and identify the two legs relative to that angle only. On elevation and depression problems, the related trap is measuring from the ground or vertical instead of the horizontal sightline.