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Trigonometric Ratios in Right Triangles · Structure and Argument · Similarity, Right Triangles, and Trigonometry · CCSS.MATH.CONTENT.HSG.SRT.C.6

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Trigonometric Ratios in Right Triangles

with Atlas

Atlas stands at the base of a tall lighthouse on a rocky coastline, holding a surveyor's transit and a sketch pad. He has measured the angle of elevation to the top of the lighthouse and the horizontal distance from his feet to its base, and is now writing side ratios in a right triangle diagram in his notepad, showing how a single angle locks in every ratio of the sides.

Identify the opposite, adjacent, and hypotenuse sides of a right triangle relative to a given acute angle.
Explain why any two right triangles sharing an acute angle must have equal sine, cosine, and tangent ratios.
Calculate sine, cosine, and tangent for an acute angle using the labeled sides of a right triangle.
Apply a trigonometric ratio equation to find an unknown side length given one side and one acute angle.
Predict which trigonometric ratio connects a given pair of sides to a known angle in a problem context.

Key terms

Hypotenuse: The longest side of a right triangle, opposite the right angle.
Opposite side: The leg directly across from the reference acute angle.
Adjacent side: The leg next to the reference acute angle, not the hypotenuse.
Sine ratio: The ratio of the opposite side to the hypotenuse for an angle.
Angle of elevation: The upward angle from the horizontal to a line of sight.

Why the Ratios Depend Only on the Angle

If two right triangles share an acute angle, they also share the right angle, so by the angle-angle criterion they are similar. Similar triangles have all corresponding sides in the same proportion, which means the quotient of any chosen pair of sides is identical across both triangles. Therefore sine, cosine, and tangent depend solely on the measure of the angle and not on the size of the triangle. This invariance is what lets a single calculator value of sin(35°) apply to every right triangle containing a 35° angle, large or small.

Labeling Sides From the Reference Angle

The labels opposite and adjacent are not fixed to corners of the page; they are assigned relative to whichever acute angle you are working with. The opposite leg is the one not touching the angle, the adjacent leg is the one forming the angle alongside the hypotenuse, and the hypotenuse is always across from the right angle. If you switch to the triangle's other acute angle, the opposite and adjacent legs swap roles while the hypotenuse stays put. Marking the reference angle before labeling prevents the most common ratio errors.

Choosing the Right Ratio to Solve

To find an unknown side, pick the ratio that links the side you want to the side you know relative to the given angle. If you know the adjacent leg and want the opposite leg, tangent connects them; if you know the hypotenuse and want the opposite leg, use sine; for the adjacent leg from the hypotenuse, use cosine. Write the ratio as an equation, isolate the unknown by multiplication, and evaluate the trigonometric value with a calculator set to degrees. This deliberate matching keeps you from forcing a formula that contains two unknowns.

Worked examples

A right triangle has a 40° angle with hypotenuse 15; find the opposite side.

The opposite side and hypotenuse are linked by sine: sin(40°) = opposite / 15.
Multiply both sides by 15: opposite = 15 · sin(40°).
Evaluate sin(40°) ≈ 0.643, so opposite ≈ 15 · 0.643 ≈ 9.64.

Answer: The opposite side is about 9.64 units.

From 20 m away, the angle of elevation to a tree top is 35°; find the tree height.

The 20 m horizontal distance is the adjacent leg and the height is the opposite leg, so use tangent.
Write tan(35°) = height / 20, then height = 20 · tan(35°).
Evaluate tan(35°) ≈ 0.700, giving height ≈ 20 · 0.700 ≈ 14.0 m.

Answer: The tree is about 14.0 m tall.

When I measured this lighthouse, I fixed my transit at one angle and watched something remarkable happen: every ratio of the triangle's sides stayed exactly the same — no matter how far back I stood or how big my sketch triangle grew. Here is why that works. Any two right triangles that share the same acute angle are similar to each other. Similar triangles have proportional sides, so the ratio opposite-over-hypotenuse is always identical for that angle, no matter the triangle's size. The same holds for the other two pairs. That's not a lucky coincidence — it is a theorem about similar figures. I write down three of these fixed ratios every time I'm in the field: • sine of angle θ = opposite ÷ hypotenuse • cosine of angle θ = adjacent ÷ hypotenuse • tangent of angle θ = opposite ÷ adjacent A compact memory hook I use: SOH-CAH-TOA. Let me show you how I used this at the lighthouse. I knew my angle of elevation θ = 35° and my horizontal distance (the adjacent leg) = 20 m, and I needed the height (opposite leg). I wrote: tan(35°) = opposite / 20, so opposite = 20 · tan(35°). My calculator gave tan(35°) ≈ 0.700, so the height ≈ 14.0 m. Critical precision I never skip: 'opposite' and 'adjacent' are always labeled relative to the angle you are using, not to some fixed corner. Before writing any ratio, mark your reference angle first, then label the sides from its perspective.

Activity

A right triangle is shown with one acute angle labeled θ = 40°, the hypotenuse marked as 15 units, and one side marked with a question mark. Drag each side-label card to the correct side, then choose the correct ratio equation and solve for the missing opposite side.

Practice

In a right triangle with a 25° angle and hypotenuse 12, find the adjacent leg using cosine.

A ladder makes a 70° angle with the ground and reaches 6 m up a wall; find the ladder's length.

Common mistakes to avoid

Bigger triangles have bigger ratiosTrigonometric ratios depend only on the angle, so similar triangles of any size share the identical sine, cosine, and tangent.
Opposite and adjacent are fixed sidesThese labels are assigned relative to the chosen reference angle and swap when you switch to the triangle's other acute angle.

Check your understanding

A right triangle has an acute angle at vertex A. The side directly across from vertex A measures 9 cm and the hypotenuse measures 19 cm. Which expression represents the sine of angle A?

Atlas is standing 30 m from the base of a lighthouse. His transit shows an angle of elevation of 28° to the top. The hypotenuse of the right triangle (from his position to the top) is 34 m. Which equation correctly finds the lighthouse height h?

A ramp rises at an angle of 20° from the ground. The ramp's length (hypotenuse) is 8 m. Which equation correctly finds the vertical height h?

Two right triangles both contain a 50° acute angle. One has a hypotenuse of 10 cm; the other has a hypotenuse of 30 cm. How do their sine ratios for the 50° angle compare?

Recap

Because right triangles sharing an acute angle are similar, the sine, cosine, and tangent ratios depend only on that angle's measure; labeling opposite and adjacent relative to the reference angle and matching the ratio that links known to unknown lets you solve for missing sides.

Reflect

How could a single measured angle help you find a height you cannot reach directly?