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Trigonometry · Lesson 2

Right-triangle ratios

Sine, cosine, and tangent begin as ratios of sides in a right triangle. Same angle, same ratios - even when the triangle scales.

Core ~18 minDeep study ~53 minWhy do the ratios only depend on the angle?What is tangent, intuitively?

Before this: Angles and turning

How this idea was born

People needed heights they could not climb: a tree across a river, a ramp's rise from its run, a tower judged from a safe distance. Surveying and astronomy kept asking the same practical question: if you can measure an angle and one length, what else can you trust?
Once turns are measurable, fix an acute angle in a right triangle and the side lengths refuse to be arbitrary. Scale the triangle up or down and the ratios stay the same, because similar triangles share angle structure. Sine, cosine, and tangent begin as names for those stable ratios.
In the fifth and sixth centuries, Aryabhata gave sine tables (half-chords) in the Aryabhatiya that made trigonometry a working computational tool in Indian astronomy. MacTutor stresses the craft: compute carefully, reuse structure, and let the sky demand better ratios. Mnemonics help memory; similarity explains why the mnemonics work.
This chapter is the triangle doorway for a modern rereader. Everything later (unit circle, waves, cosine similarity) reuses the same ratios in new costumes. Learn to label opposite, adjacent, and hypotenuse before you lean on SOH-CAH-TOA.

First principles

In a right triangle, fix an acute angle θ\theta. Name the sides relative to θ\theta:
  • Hypotenuse: side opposite the right angle (longest)
  • Opposite: side across from θ\theta
  • Adjacent: the non-hypotenuse side that touches θ\theta
θadjacentoppositehypotenuse
Fix an acute angle. Opposite, adjacent, and hypotenuse name the sides relative to that angle.
Define:
sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}
Because similar right triangles share angles, these ratios depend on θ\theta, not on how large you draw the triangle. Scale cancels in a ratio of lengths.
SOH-CAH-TOA is a memory aid:
  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent
Say the labels before you say the mnemonic. The mnemonic cannot fix a mislabeled sketch.

Worked intuition

Label before you compute

Most trig mistakes are labeling mistakes. Circle the angle. Name opposite and adjacent out loud. Only then write sine or cosine. SOH-CAH-TOA cannot save a mislabeled triangle.
Opposite and adjacent are relative to the angle you chose. Switch which acute angle you focus on, and the labels swap roles.
Labeling drill 1. In a right triangle, the right angle sits at the bottom-left corner. You care about the bottom-right acute angle. The bottom side is adjacent, the left side is opposite, the diagonal is hypotenuse.
Labeling drill 2. Same triangle, but now you care about the top acute angle. The left side becomes adjacent, the bottom side becomes opposite. Hypotenuse stays fixed.

Similarity is the explanation

SOH-CAH-TOA is a mnemonic, not an explanation. The explanation is shape families: all right triangles with a given acute angle θ\theta are scaled copies of each other. Ratios of corresponding sides are invariants of that family.
If triangle ABCABC has legs 33 and 44 and hypotenuse 55, then any similar triangle with the same acute angle has sides 6,8,106, 8, 10 or 1.5,2,2.51.5, 2, 2.5. In every case, sin⁡θ=3/5=6/10=1.5/2.5\sin\theta = 3/5 = 6/10 = 1.5/2.5.

Solving for a length

If you know one side and an acute angle, the definitions rearrange into formulas for the missing sides. Always sketch. Always label. Then rearrange:
opposite=(sin⁡θ) (hypotenuse),adjacent=(cos⁡θ) (hypotenuse)\text{opposite} = (\sin\theta)\,(\text{hypotenuse}), \quad \text{adjacent} = (\cos\theta)\,(\text{hypotenuse})
Tangent is the steepness ratio: rise over run relative to θ\theta. It can be larger than 1 when opposite exceeds adjacent (angles above 45∘45^\circ in a right triangle).
Example 1 - tree height. You stand 4040 m from a tree. The angle of elevation to the top is 35∘35^\circ. The tree height hh is opposite; your distance is adjacent. tan⁡35∘=h/40\tan 35^\circ = h/40, so h=40tan⁡35∘≈28h = 40 \tan 35^\circ \approx 28 m.
Example 2 - ladder. A 55 m ladder leans against a wall. The base is 22 m from the wall. How high does it reach? Hypotenuse =5= 5, adjacent =2= 2, so cos⁡θ=2/5\cos\theta = 2/5 and height =5sin⁡θ= 5\sin\theta. You can also use Pythagoras: height =25−4=21≈4.58= \sqrt{25-4} = \sqrt{21} \approx 4.58 m.
Example 3 - ramp. ADA guidelines often cite slope ratios. A ramp with run 1212 and rise 11 has tan⁡θ=1/12≈0.083\tan\theta = 1/12 \approx 0.083, a gentle angle. Steeper ramps mean larger tangent.
Example 4 - special angles preview. In a 45∘45^\circ-45∘45^\circ-90∘90^\circ triangle with legs 11, hypotenuse =2= \sqrt{2}. So sin⁡45∘=cos⁡45∘=1/2\sin 45^\circ = \cos 45^\circ = 1/\sqrt{2} and tan⁡45∘=1\tan 45^\circ = 1. In a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle with short leg 11, hypotenuse =2= 2, long leg =3= \sqrt{3}. Then sin⁡30∘=1/2\sin 30^\circ = 1/2, cos⁡30∘=3/2\cos 30^\circ = \sqrt{3}/2, tan⁡30∘=1/3\tan 30^\circ = 1/\sqrt{3}. The unit circle will make these numbers feel inevitable.

Inverse ratios (preview)

If sin⁡θ=0.6\sin\theta = 0.6, what is θ\theta? The inverse sine function arcsin⁡\arcsin (or sin⁡−1\sin^{-1} on calculators) answers that question. For now, know that every ratio question has two directions: angle to ratio (forward) and ratio to angle (inverse). The unit circle will make inverse trig less mysterious.

Complementary angles

In a right triangle, the two acute angles sum to 90∘90^\circ. So sin⁡θ=cos⁡(90∘−θ)\sin\theta = \cos(90^\circ - \theta). If one acute angle is 35∘35^\circ, the other is 55∘55^\circ, and sin⁡35∘=cos⁡55∘\sin 35^\circ = \cos 55^\circ. This "co-" prefix in cosine originally meant complement. Worth remembering when you check answers.

Checking answers with bounds

For any acute angle in a right triangle, 0<sin⁡θ<10 < \sin\theta < 1 and 0<cos⁡θ<10 < \cos\theta < 1. If your computation gives sin⁡θ=1.3\sin\theta = 1.3, something went wrong: labeling, calculator mode, or an inverted ratio. Bounds are cheap sanity checks before you trust a number.
Labeling drill 3: A right triangle has hypotenuse 1313 and one leg 55. Before computing any trig ratio, find the missing leg with Pythagoras (1212). Now pick either acute angle and name opposite, adjacent, and hypotenuse. Only then write sin⁡\sin, cos⁡\cos, or tan⁡\tan.
Shadow problem (similarity): A 22 m stick casts a 1.51.5 m shadow. A building casts a 4545 m shadow at the same time. Assuming parallel sun rays, the building height is 2⋅(45/1.5)=602 \cdot (45/1.5) = 60 m. The trig ratio is hiding inside the proportion.

Common confusions

Using SOH-CAH-TOA without labeling. The mnemonic is useless if "opposite" points at the wrong side.
Thinking the ratios depend on triangle size. They depend on the angle (and on being in a right triangle for these elementary definitions).
Forgetting the hypotenuse is fixed as the side opposite the right angle. Students sometimes call the longest-looking drawn segment the hypotenuse even when the right angle was mis-marked.
Expecting tangent to stay below 1. tan⁡θ>1\tan\theta > 1 whenever opposite >> adjacent for acute θ\theta.
Treating acute-triangle definitions as the whole subject. They are the doorway. The unit circle extends the same ideas to oriented angles beyond 0∘0^\circ to 90∘90^\circ.
Swapping sine and cosine. Cosine uses adjacent; sine uses opposite. Both divide by hypotenuse. If your answer is greater than 1 for an acute angle in a right triangle, you probably inverted a ratio or swapped labels.
Using the wrong angle. "Angle of elevation" is measured from horizontal up. "Angle of depression" from horizontal down. Sketch the horizontal ray first.
Calculator mode. Ensure your calculator is in the unit (degrees or radians) your problem uses. A degree problem with radian mode gives answers that look plausible but are wrong.

History and stories

In the Aryabhatiya, Aryabhata (476-550) gave sine tables (half-chords) that made trigonometry a working computational tool in Indian astronomy. MacTutor's biography emphasizes that table-building craft: compute carefully, reuse structure, let astronomy demand better ratios.
This sits in a longer chord-to-sine story that also includes Greek and later Islamic mathematicians. Do not flatten it into "Aryabhata invented sine overnight." Prefer MacTutor's trig history topic for the cross-cultural arc.
Surveying a tree with a clinometer is the same spirit outdoors: turn a measured angle into a length you could not reach with a tape.
The word "sine" traces through Latin sinus ("fold, bay") from Arabic jiba, itself from Sanskrit jya (chord-related terminology in Indian astronomy). The etymology is a reminder that trigonometry traveled across cultures before it landed in your textbook.

Real world

  • Tree or building height from a measured angle of elevation
  • Ramp rise from run (accessibility slopes, skate ramps)
  • Architecture and surveying
  • Game-engine camera pitches and simple collision geometry
Anywhere a right triangle hides inside a measurement problem, these ratios are the shortcut.
Drone photography: gimbal pitch angles control how much sky versus ground appears in frame. Small pitch changes are right-triangle steepness in disguise.
Roof pitch: builders describe slope as "rise per 12 inches run" (e.g., 4:12). That is tangent in carpentry clothing.
Lighthouse visibility: the distance to the horizon from height hh involves a right triangle with Earth's radius. Trig and Pythagoras cooperate. The geometry chapter's distance ideas meet trig ratios here.

Open when you want the machine-learning connection. Skip freely.

Field notes, extra examples, and glossary live here when you want more than the core path.

Carry this forward

  • Trig ratios are similarity invariants of an angle's right-triangle family.
  • Label opposite, adjacent, and hypotenuse before you compute.
  • Tangent measures steepness; it can exceed 1.
  • Special angles (30∘30^\circ, 45∘45^\circ, 60∘60^\circ) come from familiar triangles.
  • Next: the unit circle extends the same ideas past acute angles.

Read more

Go deeper when curiosity hits - videos, essays, and primary trails.

Check your understanding

A short learning loop - try a few, learn from misses, mark complete when you have engaged. No timer, no scoreboard.

1.sin θ equals…
2.cos θ equals…
3.tan θ equals…
4.Ratios depend only on the angle because…
5.Aryabhata's tables…

Try at least 3 core prompts, or choose I'll return later.