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Using Trig Ratios to Find Missing Sides in Right Triangles

Capítulo 4

Estimated reading time: 6 minutes

From a Question to an Equation

In most right-triangle problems, the real task is not “doing trig”; it is translating the information into one clean equation. Once you have an equation, solving is usually straightforward algebra.

A consistent solving template (use every time)

1) Identify the reference angle θ (the angle you are “using”).
2) Mark the sides relative to θ: opposite (O), adjacent (A), hypotenuse (H).
3) Choose the trig ratio that involves the sides you know and the side you want: sin uses O and H, cos uses A and H, tan uses O and A.
4) Write the equation by substituting values into the ratio.
5) Isolate the unknown using algebra (often multiplying both sides).
6) Check reasonableness: the hypotenuse must be the longest side; if 0<sin(θ)<1, then O<H; if 0<cos(θ)<1, then A<H; if tan(θ)>1, then O>A (for acute angles).

How worded descriptions map to sides

Many problems hide the triangle inside words. Translate phrases into “which side is known/unknown relative to θ.”

Wording	What it usually means
“Angle of elevation/depression”	`θ` is at the observer; vertical change is often opposite; horizontal distance is often adjacent.
“Distance along a ramp/cable/ladder”	Often the hypotenuse (a slanted length).
“Height” vs “ground distance”	Height is vertical; ground distance is horizontal. Which is opposite/adjacent depends on where `θ` is.
Diagram rotated or flipped	Rotation does not change which side is opposite/adjacent/hypotenuse relative to `θ`.

Example 1 (Sine): Solve for the Opposite Side

Problem. In a right triangle, the hypotenuse is 12 cm and the angle θ = 35°. Find the length of the side opposite θ.

Apply the template

1) Identify θ = 35°.
2) Relative sides: unknown is O, known is H = 12.
3) Pick ratio: opposite and hypotenuse suggests sin.
4) Set up equation: sin(θ) = O/H becomes sin(35°) = O/12.
5) Isolate: multiply both sides by 12: O = 12sin(35°).
6) Compute and check: O ≈ 12(0.574) ≈ 6.89 cm. Check: opposite should be less than hypotenuse (6.89 < 12), reasonable.

sin(35°) = O/12  →  O = 12 sin(35°) ≈ 6.89 cm

Example 2 (Cosine): Solve for the Adjacent Side

Problem. A right triangle has hypotenuse 20 m and angle θ = 62°. Find the adjacent side to θ.

Apply the template

Known/unknown: known H = 20, unknown A.
Pick ratio: adjacent and hypotenuse suggests cos.
Equation: cos(62°) = A/20.
Isolate: A = 20cos(62°).
Compute and check: A ≈ 20(0.469) ≈ 9.38 m. Check: adjacent should be less than hypotenuse (9.38 < 20), reasonable.

cos(62°) = A/20  →  A = 20 cos(62°) ≈ 9.38 m

Example 3 (Tangent): Solve Using Opposite and Adjacent

Problem. In a right triangle, the adjacent side to θ is 7.5 ft and θ = 28°. Find the opposite side.

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Apply the template

Known/unknown: known A = 7.5, unknown O.
Pick ratio: opposite and adjacent suggests tan.
Equation: tan(28°) = O/7.5.
Isolate: O = 7.5tan(28°).
Compute and check: O ≈ 7.5(0.532) ≈ 3.99 ft. Check: since tan(28°)<1, opposite should be smaller than adjacent (3.99 < 7.5), reasonable.

tan(28°) = O/7.5  →  O = 7.5 tan(28°) ≈ 3.99 ft

Common Algebra Moves (So You Don’t Get Stuck)

Most “solve for a side” equations look like one of these. Practice the algebra pattern so trig problems feel routine.

Setup	Isolate the unknown
`sin(θ) = O/H` and `H` is known	`O = H sin(θ)`
`cos(θ) = A/H` and `H` is known	`A = H cos(θ)`
`tan(θ) = O/A` and `A` is known	`O = A tan(θ)`
`sin(θ) = O/H` and `O` is known	`H = O / sin(θ)`
`cos(θ) = A/H` and `A` is known	`H = A / cos(θ)`
`tan(θ) = O/A` and `O` is known	`A = O / tan(θ)`

Reasonableness Checks You Should Always Do

Hypotenuse check: if you solved for H, it must be larger than both legs.
Size vs ratio: for acute angles, 0<sin(θ)<1 and 0<cos(θ)<1. So multiplying by sin or cos should make a number smaller; dividing by them should make a number larger.
Tangent comparison: if θ is small, tan(θ) is small, so opposite should be noticeably smaller than adjacent. If θ is close to 90°, tan(θ) is large, so opposite should be much larger than adjacent.
Orientation doesn’t matter: a triangle drawn “leaning left” or “upside down” still uses the same relative-side logic to θ.

Practice Set A (Fully Scaffolded)

For each problem: (1) identify θ, (2) name known sides relative to θ, (3) choose ratio, (4) write equation, (5) solve, (6) check reasonableness.

Sine (find opposite). θ = 41°, hypotenuse 15 m. Find O.
Hint: sin(41°) = O/15.
Cosine (find adjacent). θ = 18°, hypotenuse 9.2 cm. Find A.
Hint: cos(18°) = A/9.2.
Tangent (find opposite). θ = 53°, adjacent 6 ft. Find O.
Hint: tan(53°) = O/6.
Tangent (find adjacent). θ = 35°, opposite 4.8 in. Find A.
Hint: tan(35°) = 4.8/A.

Practice Set B (Less Scaffolded)

A right triangle has θ = 67° and hypotenuse 24 units. Find the leg adjacent to θ.
A right triangle has θ = 12° and adjacent side 30 m. Find the opposite side.
A ramp makes a 9° angle with the ground. The ramp length is 8.0 m. How high does it rise vertically? (Assume the ramp and ground form a right triangle.)
A cable is attached to the top of a pole. The cable makes a 58° angle with the ground and the ground distance from the pole to the anchor point is 11 m. Find the cable length.

Practice Set C (Minimal Scaffolding + Different Orientations)

These are intentionally drawn/imagined in varied orientations. The math is the same; focus on which side is opposite/adjacent to the given θ.

A right triangle is drawn with the right angle at the top. The side opposite the right angle is labeled H. Angle θ is at the bottom-left vertex and equals 39°. The hypotenuse is 17. Find the side opposite θ.
A right triangle is rotated so the hypotenuse slopes down from left to right. Angle θ = 74° is at the rightmost vertex. The side adjacent to θ (not the hypotenuse) is 5.5. Find the hypotenuse.
A right triangle has θ = 46°. The opposite side is 9. Find the adjacent side.
A right triangle has hypotenuse 13 and angle θ = 25°. Find the adjacent side, then state whether your answer makes sense compared to the hypotenuse.
A right triangle has adjacent side 4 and opposite side 10 relative to θ. Without finding θ, decide whether θ is closer to 0° or to 90°, and justify using tangent.

Now answer the exercise about the content:

A right triangle has angle θ = 74° and the adjacent side to θ (not the hypotenuse) is 5.5. Which equation correctly sets up finding the hypotenuse H?

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You missed! Try again.

Cosine relates adjacent and hypotenuse: cos(θ)=A/H. With A=5.5 and θ=74°, the correct setup is cos(74°)=5.5/H, then solve by dividing 5.5 by cos(74°).