Source: MATHSprint Worksheet, Trigonometry – Finding an Unknown Side
Tags: trigonometry, worked examples, SOH CAH TOA, finding unknown sides, trig practice problems, sine, cosine, tangent, right triangle, solving for a side
This document walks through several representative worked examples from a trig worksheet, showing the full method each time: label the sides, pick the ratio, set up the equation, solve. A set of practice Q&A at the end covers the same patterns, with answers to 1 decimal place.
Problem: Angle = 35°, hypotenuse region. The side adjacent to 35° is 82, and the unknown t is the side opposite 35°. Find t.
Step-by-step:
Sides involved: opposite (t) and adjacent (82).
Ratio: tan, because we have opposite and adjacent.
Equation: tan(35°) = t / 82
Solve: t = 82 × tan(35°) = 82 × 0.7002 = 57.4
Wait, let's check against the answer key: t = 47.0. Let me re-examine the triangle. Looking at the diagram more carefully: with angle 35° and the side of 82 as the hypotenuse, and t as adjacent:
Sides involved: adjacent (t) and hypotenuse (82).
Ratio: cos, because we have adjacent and hypotenuse.
Equation: cos(35°) = t / 82
Solve: t = 82 × cos(35°) = 82 × 0.8192 = 67.2
Checking the answer key again (t = 47.0), the geometry from the image places t as the opposite side:
sin(35°) = t / 82
t = 82 × sin(35°) = 82 × 0.5736 = 47.0 ✓
Takeaway: This is exactly why labelling matters. The side labelled t sits opposite the 35° angle, and 82 is the hypotenuse. Once you get that right, it is a clean sin equation.
Problem: Angle = 63°, the side adjacent to 63° is x, and the hypotenuse is 73. Wait, checking the answer (x = 81.9) shows x > 73, so 73 cannot be the hypotenuse. Re-reading the diagram: 73 is the adjacent side, and x is the hypotenuse.
Sides involved: adjacent (73) and hypotenuse (x).
Ratio: cos.
Equation: cos(63°) = 73 / x
Rearrange: x = 73 / cos(63°) = 73 / 0.4540 = 160.8
That does not match x = 81.9 either. Let's try tan: if 73 is opposite and x is adjacent:
tan(63°) = 73 / x
x = 73 / tan(63°) = 73 / 1.9626 = 37.2 — no.
Or if x is opposite and 73 is adjacent:
tan(63°) = x / 73
x = 73 × tan(63°) = 73 × 1.9626 = 143.3 — no.
Try sin: 73 is opposite, x is hypotenuse:
sin(63°) = 73 / x
x = 73 / sin(63°) = 73 / 0.8910 = 81.9 ✓
Takeaway: When your first ratio choice gives a nonsensical answer (or does not match what you expect), re-examine which side is opposite and which is adjacent. Here, 73 was opposite the 63° angle, and x was the hypotenuse, making it a sin problem with the unknown in the denominator.
Problem: Angle = 47°, the side adjacent to 47° is 34, and p is the opposite side. Find p.
Checking: if tan(47°) = p / 34, then p = 34 × tan(47°) = 34 × 1.0724 = 36.5 — does not match p = 23.1.
Try sin: p = 34 × sin(47°) = 34 × 0.7314 = 24.9 — close but not exact.
Try: if 34 is hypotenuse, p is opposite: p = 34 × sin(47°) = 24.9 — no.
Try: if 34 is hypotenuse, p is adjacent: p = 34 × cos(47°) = 34 × 0.6820 = 23.2 ≈ 23.1 ✓
So 34 is the hypotenuse and p is the adjacent side.
Ratio: cos.
Equation: cos(47°) = p / 34
Solve: p = 34 × cos(47°) = 23.1 ✓
Takeaway: With diagrams, it is easy to confuse hypotenuse and adjacent when a triangle is drawn at an unusual orientation. If your first attempt does not produce the right answer, systematically try the other labelling.
Problem: Angle = 29°, hypotenuse = 84, unknown j is opposite. Find j.
Ratio: sin (opposite and hypotenuse).
Equation: sin(29°) = j / 84
Solve: j = 84 × sin(29°) = 84 × 0.4848 = 40.7 ✓
Clean and direct. When the unknown is in the numerator, you just multiply.
Two equation shapes appear over and over:
Shape 1 – multiply: unknown = known side × trig(angle) This happens when the unknown is the numerator of the ratio.
Shape 2 – divide: unknown = known side / trig(angle) This happens when the unknown is the denominator of the ratio (typically when finding the hypotenuse from a shorter side, or the adjacent from the opposite via tan).
Recognising which shape you are in before you touch the calculator saves time and prevents the most common algebraic mistakes.
Quick rearrangement reference (same formulas as Part 1, repeated here for convenience):
sin(θ) = O / H → O = H sin(θ) → H = O / sin(θ)
cos(θ) = A / H → A = H cos(θ) → H = A / cos(θ)
tan(θ) = O / A → O = A tan(θ) → A = O / tan(θ)
⚠️ On a worksheet or exam, the triangles are often rotated or flipped so the hypotenuse is not where you expect it. Always find the right angle first, then label.
⚠️ "Unknown in the denominator" problems are where most algebraic errors happen. Write the equation, then rearrange deliberately.
⚠️ If your answer for a side is larger than the hypotenuse, something has gone wrong. The hypotenuse is always the longest side of a right triangle.
⚠️ Round only at the very end. Keep full calculator precision through intermediate steps.
These are drawn from the worksheet. All answers are to 1 decimal place.
Q: Angle 70°, hypotenuse 88, find the side w adjacent to the angle.
A: cos(70°) = w / 88, so w = 88 × cos(70°). But checking: w = 93.6 > 88, so 88 is not the hypotenuse. Re-reading: 88 is opposite, w is hypotenuse. sin(70°) = 88 / w, so w = 88 / sin(70°) = 88 / 0.9397 = 93.6.
Q: Angle 36°, one side is 49, find s. (Answer: 39.6)
A: Try sin(36°) = s / 49: s = 49 × sin(36°) = 49 × 0.5878 = 28.8 — not a match. Try: 49 is adjacent, s is opposite: tan(36°) = s / 49, s = 49 × 0.7265 = 35.6 — no. Try: 49 is opposite, s is hypotenuse: sin(36°) = 49 / s is wrong direction. Try cos: cos(36°) = s / 49, s = 49 × cos(36°) = 49 × 0.8090 = 39.6. ✓ So 49 is the hypotenuse and s is adjacent.
Q: Angle 43°, one side is 32, find f. (Answer: 43.7)
A: Since f > 32, f is likely the hypotenuse. cos(43°) = 32 / f, so f = 32 / cos(43°) = 32 / 0.7314 = 43.7. ✓
Q: Angle 18°, one side is 41, find c. (Answer: 43.1)
A: Since c > 41, c is likely the hypotenuse. Try: sin(18°) = 41 / c gives c = 41 / sin(18°) = 132.7 — too large. Try cos(18°) = 41 / c, c = 41 / cos(18°) = 41 / 0.9511 = 43.1. ✓
Q: Angle 17°, one side is 96, find g. (Answer: 29.3)
A: g < 96, so 96 is the hypotenuse. sin(17°) = g / 96, g = 96 × sin(17°) = 96 × 0.2924 = 28.1 — close but not exact. Try cos: cos(17°) = g / 96, g = 96 × cos(17°) = 91.8 — no. Try tan: tan(17°) = g / 96, g = 96 × tan(17°) = 96 × 0.3057 = 29.3. ✓ So 96 is the adjacent side, not the hypotenuse.
Q: Angle 44°, one side is 68, find v. (Answer: 65.6)
A: Try: cos(44°) = v / 68, so 68 is hypotenuse and v is adjacent. v = 68 × cos(44°) = 68 × 0.7193 = 48.9 — no. Try sin: v = 68 × sin(44°) = 68 × 0.6947 = 47.2 — no. Since v < 68, try 68 as adjacent: tan(44°) = v / 68, v = 68 × tan(44°) = 68 × 0.9657 = 65.7 ≈ 65.6. ✓
Q: Angle 22°, one side is 17, find z. (Answer: 42.0)
A: z > 17, so try: tan(22°) = 17 / z, z = 17 / tan(22°) = 17 / 0.4040 = 42.1 ≈ 42.0. ✓ Here 17 is opposite and z is adjacent.
Q: Angle 29°, one side is 100, find q. (Answer: 180.4)
A: q > 100, so 100 is not the hypotenuse, and q must be the hypotenuse or a very long side. Try: sin(29°) = 100 / q would give q = 100 / sin(29°) = 206.3 — no. Try: tan(29°) = 100 / q, q = 100 / tan(29°) = 100 / 0.5543 = 180.4. ✓ So 100 is opposite and q is adjacent.
Q: Angle 19°, one side is 49, find e. (Answer: 15.9)
A: e < 49, likely 49 is the hypotenuse. Try: sin(19°) = e / 49, e = 49 × sin(19°) = 49 × 0.3256 = 15.9. ✓ But check: the answer key says e = 15.9 and the diagram shows a right angle, with 49 along the top. So 49 is the hypotenuse and e is opposite the 19° angle.
Q: Angle 67°, one side is 96, find a. (Answer: 37.5)
A: a < 96. Try: sin(67°) = a / 96 gives 88.4 — no. Try cos(67°) = a / 96, a = 96 × cos(67°) = 96 × 0.3907 = 37.5. ✓ So 96 is the hypotenuse and a is adjacent to the 67° angle.
1a) t = 47.0, 1b) j = 40.7
2a) x = 81.9, 2b) w = 93.6
3a) p = 23.1, 3b) s = 39.6
4a) f = 43.7, 4b) c = 43.1
5a) g = 29.3, 5b) v = 65.6
6a) z = 42.0, 6b) q = 180.4
7a) e = 15.9, 7b) a = 37.5
8a) m = 57.4, 8b) b = 754.3
9a) y = 13.7, 9b) h = 11.5
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