Source: MATHSprint Worksheet, Trigonometry – Finding an Unknown Side
Tags: trigonometry, trig ratios, SOH CAH TOA, SOHCAHTOA, finding unknown sides, right triangle, sine, cosine, tangent, opposite, adjacent, hypotenuse, solving triangles
Right-triangle trigonometry lets you find a missing side when you know one side and one acute angle. You pick the correct ratio (sin, cos, or tan) based on which sides are involved relative to the given angle, write the equation, and solve for the unknown. The mnemonic SOH CAH TOA keeps the three ratios straight.
Hypotenuse
The longest side of a right triangle, always opposite the right angle.
Opposite side
The side directly across from the angle you are working with. Changes depending on which angle you pick.
Adjacent side
The side next to the angle you are working with (that is not the hypotenuse). Also changes depending on which angle you pick.
Sine (sin)
The ratio of the opposite side to the hypotenuse. sin(θ) = opposite / hypotenuse
Cosine (cos)
The ratio of the adjacent side to the hypotenuse. cos(θ) = adjacent / hypotenuse
Tangent (tan)
The ratio of the opposite side to the adjacent side. tan(θ) = opposite / adjacent
SOH CAH TOA
A mnemonic for remembering the three trig ratios:
Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent
Before choosing a ratio, label all three sides from the perspective of the acute angle you have been given:
Look at the right angle to find the hypotenuse (always the side opposite the 90° angle).
The side directly across from the given acute angle is the opposite.
The remaining side (next to the given angle, not the hypotenuse) is the adjacent.
These labels shift if you switch to the other acute angle. Always label relative to the angle stated in the problem.
Once the sides are labelled, identify which two sides are involved: the one you know and the one you need.
Opposite and hypotenuse → use sin
Adjacent and hypotenuse → use cos
Opposite and adjacent → use tan
A quick way to remember: cross out the side that is neither known nor wanted, and the remaining pair tells you which ratio to use.
Write the ratio equation with the angle and the two relevant sides.
Substitute the known values.
Rearrange to isolate the unknown side.
Two patterns come up repeatedly:
When the unknown is in the numerator (e.g. opposite = hypotenuse × sin(θ)): Multiply both sides by the known denominator.
When the unknown is in the denominator (e.g. hypotenuse = opposite / sin(θ)): Rearrange so the unknown moves to one side. This usually means dividing the known side by the trig value.
Make sure your calculator is set to degrees, not radians. This is the single most common source of wrong answers on worksheets and exams.
The three core formulas, rearranged for every unknown:
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)
Where O = opposite, A = adjacent, H = hypotenuse, θ = given acute angle.
⚠️ The most tested skill: correctly identifying opposite, adjacent, and hypotenuse relative to the given angle. Mislabelling the sides is the number-one error.
⚠️ When the unknown is the denominator of the ratio, students often multiply instead of dividing (or vice versa). Always write the ratio equation first, then rearrange. Do not skip steps.
⚠️ Calculator in radians mode will produce wildly wrong answers. Check before every exam.
⚠️ Answers are typically required to 1 decimal place. Round only at the final step; rounding mid-calculation introduces error.
⚠️ Trigonometric ratios only apply to right triangles. If there is no right angle, you need the sine rule or cosine rule instead.
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