To find sin, cos and tan of a triangle, label the three sides first — opposite, adjacent and hypotenuse — then use SOH-CAH-TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. For a 3-4-5 triangle with θ opposite the side 3, that gives sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8 and tan θ = 3/4 = 0.75 — not 4/5.
Sin, cos and tan look mysterious until you see what they really are: three ratios of a right triangle’s sides. They are not three separate facts to memorise — they are three fractions made from the same three numbers. Once the triangle is labelled, each ratio is one division. The rest of this post is about labelling correctly, because that is where almost every wrong answer comes from.
Check any value instantly with a free trigonometry calculator, which shows the working step by step.
Step 1: Label the Triangle — Opposite, Adjacent, Hypotenuse
Every right triangle has three sides, and their names depend on which angle you are working with. Call that angle θ (theta).
- Hypotenuse: the longest side, always opposite the 90° angle. It never changes its name, whichever angle you pick.
- Opposite: the side across the triangle from θ — the one that does not touch θ at all.
- Adjacent: the side next to θ that is not the hypotenuse — the one that touches θ and the right angle.
Take the 3-4-5 triangle and put θ at the corner opposite the side 3. Then opposite = 3, adjacent = 4 and hypotenuse = 5. Now move θ to the other acute corner — the one opposite the side 4. Suddenly opposite = 4 and adjacent = 3, while the hypotenuse stays 5. The sides did not move; only the labels did, because the labels follow θ.
The rule to never forget: label the triangle for the angle in your question first, before you write any ratio. If you label for the wrong angle, opposite and adjacent swap — and every answer comes out wrong.
Step 2: SOH-CAH-TOA — The Three Ratios
SOH-CAH-TOA packs all three definitions into one word:
- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent
Say it while you point at each side and the formula writes itself. There is also a shortcut worth knowing: tan θ = sin θ / cos θ. Divide the first two ratios and the hypotenuses cancel, leaving opposite/adjacent — exactly the definition of tan. That is why tan is sometimes called “sin over cos”.
Worked example: the 3-4-5 triangle. With θ opposite the side 3 (opposite = 3, adjacent = 4, hypotenuse = 5):
- sin θ = 3/5 = 0.6
- cos θ = 4/5 = 0.8
- tan θ = 3/4 = 0.75
Check with the identity: sin²θ + cos²θ = 0.6² + 0.8² = 0.36 + 0.64 = 1. This identity holds for every angle, so it is the fastest way to check your three ratios: if the squares do not add to 1, at least one ratio is wrong.
The 3-4-5 Trap: Why tan θ Is 3/4, Not 4/5
Here is the question that catches most students: if sin θ = 3/5, what is tan θ? The tempting answer is 4/5 — it uses the two numbers you have not used yet, and it feels right. It is wrong.
Tangent is opposite over adjacent, and the adjacent side is not 5 — 5 is the hypotenuse. You have to find the adjacent side first, with Pythagoras: adjacent² = hypotenuse² − opposite² = 5² − 3² = 25 − 9 = 16, so adjacent = 4. Then tan θ = opposite/adjacent = 3/4 = 0.75. The number 4/5 is cos θ, not tan θ — the trap is mixing the two up.
The shortcut habit: whenever you see sin θ = 3/5, think “3-4-5” immediately. The missing side is 4, the angle is 36.87°, and tan θ is 3/4. The same reflex works for the other Pythagorean triples you will meet: 5-12-13 and 8-15-17.
Worked Example: The 5-12-13 Triangle
Take a right triangle with opposite 5 and adjacent 12, and find sin, cos and tan.
Step 1 — find the hypotenuse. Pythagoras: hypotenuse² = 5² + 12² = 25 + 144 = 169, so hypotenuse = 13. (If you spotted 5-12-13, you already knew.)
Step 2 — write the ratios.
- sin θ = 5/13 = 0.3846
- cos θ = 12/13 = 0.9231
- tan θ = 5/12 = 0.4167
Step 3 — check. sin²θ + cos²θ = 0.3846² + 0.9231² = 0.1479 + 0.8521 = 1 ✓ (to rounding). And tan θ = sin θ/cos θ = 0.3846/0.9231 = 0.4167 ✓ — the two routes agree.
Step 4 — the angle, if you need it. θ = arcsin(0.3846) = 22.62°. The other acute angle is 90° − 22.62° = 67.38°. To check your work, see how to find sin cos tan on the free calculator — it returns the ratios, hypotenuse and both angles from just the two sides.
Worked Example: The 8-15-17 Triangle
Same recipe, different triple. Opposite 8, adjacent 15.
Hypotenuse: √(8² + 15²) = √(64 + 225) = √289 = 17.
Ratios:
- sin θ = 8/17 = 0.4706
- cos θ = 15/17 = 0.8824
- tan θ = 8/15 = 0.5333
Angle: θ = arcsin(8/17) = 28.07°, and the other angle is 90° − 28.07° = 61.93°.
Notice the pattern across all three triples: the ratios are just the side lengths divided by each other. There is no magic — sin, cos and tan are fractions, and the only skill is knowing which sides go in which fraction.
What If You Only Know Two Sides?
Sometimes the problem gives you two sides but not the ones you need. The fix is always Pythagoras first, ratios second.
Example: opposite 6, hypotenuse 10. You want tan θ, but tan needs the adjacent side. Find it: adjacent² = 10² − 6² = 100 − 36 = 64, so adjacent = 8. Now sin θ = 6/10 = 0.6, cos θ = 8/10 = 0.8, tan θ = 6/8 = 0.75 — the 3-4-5 triangle again, doubled. The angle is arcsin(0.6) = 36.87°.
Example: adjacent 9, hypotenuse 15. The opposite side is √(15² − 9²) = √(225 − 81) = √144 = 12. So sin θ = 12/15 = 0.8, cos θ = 9/15 = 0.6, tan θ = 12/9 = 1.3333. The angle is arccos(0.6) = 53.13° — the other acute angle of the 3-4-5 triangle.
The scaling insight: 6-8-10 gives exactly the same ratios as 3-4-5, because every side doubled and the ratios cancelled the doubling. Sin, cos and tan depend only on the angle, never on the size of the triangle. That is why the same sin 30° = 0.5 works for a triangle the size of a coin and one the size of a football field.
Finding the Angle: Inverse Trig
So far the angle was given or found from the sides. The reverse question is just as common: you know a ratio, find the angle. That is what arcsin, arccos and arctan do — they run the functions backwards.
- If sin θ = 0.5, then θ = arcsin(0.5) = 30°.
- If cos θ = 0.5, then θ = arccos(0.5) = 60°.
- If tan θ = 1, then θ = arctan(1) = 45°.
- If sin θ = 0.6 (the 3-4-5 triangle), then θ = arcsin(0.6) = 36.87°.
The range rule: sin θ and cos θ can never be bigger than 1, so arcsin(2) has no answer — if your ratio comes out above 1, the sides are mislabelled, usually because the hypotenuse was entered as the opposite side. Arctan has no such limit: tan θ can be any number, so arctan accepts anything.
Real-World Example: The Ladder Problem
A 10-foot ladder leans against a wall at 70° to the ground. How high does it reach, and how far is its foot from the wall?
Draw the triangle: the ladder is the hypotenuse (10), the height up the wall is the opposite side, and the distance along the ground is the adjacent side.
- Height = 10 × sin 70° = 10 × 0.9397 = 9.3969 ft
- Base distance = 10 × cos 70° = 10 × 0.3420 = 3.4202 ft
Check with Pythagoras: 9.3969² + 3.4202² = 88.30 + 11.70 = 100 = 10² ✓ — the ladder length comes back, so the sides are right.
Real-World Example: The Shadow Problem
A building is 20 units tall, and the sun casts a shadow at an angle of elevation of 30°. How long is the shadow?
Here you know the opposite side (the building, 20) and the angle (30°), and you want the adjacent side (the shadow). Tan links opposite and adjacent: tan 30° = 20/shadow, so shadow = 20/tan 30° = 20/0.5774 = 34.6410 units. The shadow is longer than the building is tall — which makes sense, because the sun is low at 30°.
How to Find Sin Cos Tan: The 6-Step Checklist
- Find the right angle and mark the hypotenuse — the side opposite it, always the longest.
- Mark θ — the angle in your question — then label the side across from it as opposite and the side next to it as adjacent.
- Find any missing side with Pythagoras before you write a ratio (a² + b² = c²).
- Write the ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Check with sin²θ + cos²θ = 1 — if it does not add to 1, re-label.
- Find the angle with inverse trig if the question asks for it: θ = arcsin, arccos or arctan of the ratio.
Common Mistakes When Finding Sin, Cos and Tan
Using the hypotenuse in tan. Tan is opposite over adjacent — the hypotenuse never appears in it. If your tan uses the longest side, it is wrong.
Labelling from the wrong angle. Opposite and adjacent swap when θ moves to the other corner. In a 3-4-5 triangle, sin θ is 0.6 from one corner and 0.8 from the other — both are correct, for different angles. Know which angle the question means.
Degrees vs radians. Most calculators default to radians. Sin(30°) = 0.5, but sin(30 radians) = −0.9880. If your answer looks nothing like the ratios above, check the angle mode first.
Forgetting tan 90° is undefined. Tan 90° = 1/0, division by zero. It is not 0 and not 1 — it has no value, and a calculator showing an enormous number is saying exactly that.
Skipping the check. The identity sin²θ + cos²θ = 1 takes five seconds and catches almost every labelling error. Use it every time.
Frequently Asked Questions
How do you find sin cos tan of a triangle?
Label the sides, then use SOH-CAH-TOA. Opposite is across from θ, adjacent is next to θ, hypotenuse is opposite the right angle. Then sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. For 3-4-5 with θ opposite the 3: 0.6, 0.8 and 0.75.
If sin θ = 3/5, what is tan θ?
3/4 = 0.75. Find the adjacent side first with Pythagoras: 5² − 3² = 16, so adjacent = 4, and tan θ = 3/4. The answer is not 4/5 — that is cos θ.
What is sin cos tan of 30 degrees?
Sin 30° = 0.5, cos 30° = 0.8660, tan 30° = 0.5774. These are the exact values 1/2, √3/2 and 1/√3 from the 30-60-90 triangle.
How do you find the angle from sin cos tan?
Use inverse trig: θ = arcsin, arccos or arctan of the ratio. Arcsin(0.5) = 30°, arccos(0.5) = 60°, arctan(1) = 45°. The ratio for arcsin and arccos must be between −1 and 1.
Do sin cos tan change if the triangle is bigger?
No. The ratios depend only on the angle. A 6-8-10 triangle has exactly the same sin, cos and tan as a 3-4-5 triangle — 0.6, 0.8 and 0.75 — because scaling all sides by the same factor cancels out in the fractions.
What is the easiest way to remember sin cos tan?
SOH-CAH-TOA. Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Label the triangle first, then read the ratio off the word.
Try It Yourself
The fastest way to make this stick is to run the numbers yourself: enter any two sides of a right triangle into a free trigonometry calculator and compare its sin, cos and tan with your hand calculation — including the 3-4-5 trap (tan θ = 3/4, not 4/5). If you need the missing side first, the Pythagorean Theorem Calculator finds it in one step, and the Triangle Calculator solves non-right triangles with the law of sines and cosines.