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Right-Triangle Trigonometry
How to measure a tree you can’t climb.
A surveyor wants the height of a pine tree. She can’t climb it and she can’t drop a tape measure from the top. What she can do is pace out 50 meters from the trunk, point an instrument at the treetop, and read one angle. Those two facts, one distance and one angle, are enough to work out the height.
The tool that makes this work is trigonometry: the math that links a triangle’s angles to its side lengths. By the end of this unit you’ll do the surveyor’s calculation yourself.
Grades 9–12 Unit 1 of Trigonometry and Pre-Calculus
Step 1Naming the sides of a right triangle
A right triangle is a triangle with one 90° angle, called the right angle, often marked with a small square. Its other two angles are acute, meaning smaller than 90°, and together they add up to 90°.
We pick one of the acute angles and call it θ (the Greek letter theta, a common name for an angle). Then each side gets a name relative to θ:
Hypotenuse: the longest side, across from the right angle. It never changes.
Opposite: the leg across from θ. It doesn’t touch θ at all.
Adjacent: the leg that touches θ and isn’t the hypotenuse. (Adjacent means “next to.”)
Tip: find the hypotenuse first (across from the square corner). Then put your finger on θ. The leg your finger touches is adjacent; the leg it can’t reach is opposite.
Step 2The three ratios: SOH-CAH-TOA
A ratio is one number divided by another. Trigonometry is built on three ratios of side lengths, each named after a trig function of the angle θ: sine (written sin), cosine (cos) and tangent (tan).
sin θ = oppositehypotenuse
cos θ = adjacenthypotenuse
tan θ = oppositeadjacent
Read SOH-CAH-TOA as “Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.”
Why do these ratios only depend on the angle?
Take two right triangles that both have a 35° angle. Their third angles are both 55°, so they have exactly the same shape. Triangles with the same angles are called similar: one is just a scaled-up copy of the other. If you double every side, each ratio becomes (2 × top) ÷ (2 × bottom), and the 2s cancel. So sin 35° is the same number for every right triangle with a 35° angle. Your calculator simply has these numbers built in.
Example: the 3-4-5 triangle
A right triangle has legs 3 and 4 and hypotenuse 5 (check: 3² + 4² = 9 + 16 = 25 = 5²). Let θ be the angle across from the side of length 3. Then the opposite side is 3 and the adjacent side is 4.
Check: tan θ should equal sin θ ÷ cos θ, and 0.6 ÷ 0.8 = 0.75. ✓
Notice that sine and cosine of an acute angle are always between 0 and 1, because a leg is always shorter than the hypotenuse. Tangent can be any positive number.
Try itTriangle Explorer
Step 3Special right triangles and exact values
Most trig values are endless decimals, so calculators round them. But two triangles give exact values, written with square roots instead of decimals. You can build both from shapes you already know.
45-45-90: half a square
Cut a square with side 1 along its diagonal. The legs are both 1, and by the Pythagorean theorem the hypotenuse is √(1² + 1²) = √2. Sides: 1 : 1 : √2.
30-60-90: half an equilateral triangle
Cut an equilateral triangle with side 2 down the middle. The short leg is 1, the hypotenuse is 2, and the long leg is √(2² − 1²) = √3. Sides: 1 : √3 : 2.
Reading SOH-CAH-TOA off those triangles gives the exact values. (We rationalize fractions like 1/√2 by multiplying top and bottom by √2, giving √2/2.)
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Look for patterns: sin 30° = cos 60° and sin 60° = cos 30°. That’s because the 30° and 60° angles sit in the same triangle, so the side opposite one is adjacent to the other.
Example: scaling the special triangles
A 45-45-90 triangle has legs of 7 cm. Multiply every side of 1 : 1 : √2 by 7:
A 30-60-90 triangle has a short leg of 7 cm. Multiply 1 : √3 : 2 by 7:
Check with Pythagoras: 7² + (7√3)² = 49 + 147 = 196 = 14². ✓
Step 4Finding a missing side
If you know one acute angle and one side, you can find any other side. The recipe:
- Label the sides opposite, adjacent and hypotenuse relative to the known angle.
- Pick the ratio that uses the side you know and the side you want.
- Write the equation and solve it. Round only at the end.
Calculator check: this unit measures angles in degrees. Make sure your calculator is in degree mode (often shown as DEG or D). Test it: sin 30 should give 0.5. If you get −0.988, it’s in radian mode.
Example 1: hypotenuse 12, angle 35°
Find the legs x (adjacent to 35°) and y (opposite 35°), to the nearest hundredth.
Check with Pythagoras: 6.88² + 9.83² ≈ 47.33 + 96.63 = 143.96 ≈ 144 = 12². ✓
Example 2: adjacent side 15, angle 28°
Find the hypotenuse h. This time the unknown is on the bottom of the fraction, so it takes one extra step.
For the opposite side: tan 28° = y/15, so y = 15 tan 28° ≈ 7.98.
Check: 15² + 7.98² ≈ 225 + 63.68 = 288.68, and √288.68 ≈ 16.99. ✓
Step 5Finding a missing angle with inverse trig
Now flip the problem: you know two sides and want the angle. You need to undo sine, cosine or tangent. The functions that do that are the inverse trig functions, written sin⁻¹, cos⁻¹ and tan⁻¹ (also called arcsin, arccos and arctan).
If sin θ = 0.7, then θ = sin⁻¹(0.7): “the angle whose sine is 0.7.”
Watch out: the −1 in sin⁻¹ is not an exponent. sin⁻¹(0.7) does not mean 1 ÷ sin(0.7). On most calculators you press 2nd or SHIFT, then sin.
Example 3: opposite 7, hypotenuse 10
The other acute angle is 90° − 44.4° = 45.6°.
Check: sin 44.4° ≈ 0.6997, and 0.6997 × 10 ≈ 7.0. ✓
Example 4: legs 5 and 8
The side of length 5 is opposite θ and the side of length 8 is adjacent to it.
The other acute angle is 90° − 32.0° = 58.0°, and the hypotenuse is √(5² + 8²) = √89 ≈ 9.43.
Check: tan 32.0° ≈ 0.6249 ≈ 0.625. ✓
Step 6Angles of elevation and depression
Real problems describe angles from someone’s point of view, always measured from a horizontal (level) line.
Angle of elevation: the angle you tilt your line of sight up from horizontal to see something above you.
Angle of depression: the angle you tilt your line of sight down from horizontal to see something below you.
Why are they equal? The line of sight crosses two parallel horizontal lines, and the two angles are alternate interior angles, which are always equal. So you can always put the angle at the bottom of your triangle, where it’s easier to work with.
Example 5: the surveyor’s tree
The surveyor from the start stands 50 m from the tree. Her instrument is 1.5 m above the ground, and the angle of elevation to the top of the tree is 38°. How tall is the tree?
Check: 39.06 ÷ 50 = 0.7812, and tan⁻¹(0.7812) ≈ 38.0°. ✓
Example 6: the lighthouse and the boat
From the top of a 30 m lighthouse, a keeper sees a boat at an angle of depression of 12°. How far is the boat from the base of the lighthouse?
Check: 30 ÷ 141.1 ≈ 0.2126 = tan 12°. ✓ The boat is about 141 m out.
Step 7The reciprocal ratios: csc, sec and cot
Flip each of the three main ratios upside down and you get three more. A reciprocal is 1 divided by a number, so the reciprocal of 3/5 is 5/3.
| Name | Definition | Sides | 3-4-5 example |
|---|---|---|---|
| cosecant (csc) | 1 ÷ sin θ | hyp ÷ opp | 5/3 |
| secant (sec) | 1 ÷ cos θ | hyp ÷ adj | 5/4 |
| cotangent (cot) | 1 ÷ tan θ | adj ÷ opp | 4/3 |
A memory trick: each pair has exactly one “co”: sin ↔ cosecant, cos ↔ secant, tan ↔ cotangent.
Don’t mix them up: csc θ = 1/sin θ, but sin⁻¹ is the inverse function that gives back an angle. Most calculators have no csc key; type 1 ÷ sin instead.
HistoryWhere did “sine” come from?
The Greek astronomer Hipparchus of Nicaea (about 190–120 BCE) built what was probably the first trigonometric table, a table of chords (lines across a circle) for working out the positions of the Sun and Moon. The historian Thomas Heath called him the first person whose systematic use of trigonometry we have written evidence for.
Around 500 CE, the Indian mathematician Aryabhata used half-chords instead, which is our modern sine. He called it jya. Arabic scholars wrote the word as jiba, and because Arabic is often written without vowels it was later read as jaib, meaning “fold” or “bay.” Twelfth-century Latin translators turned jaib into sinus, the Latin for the same thing, and that became our word sine.
Source: MacTutor History of Mathematics, University of St Andrews (links below).
Your turnPractice Generator
Check yourselfRight-Triangle Quiz
Ten questions. Choose one answer for each, then press Check my answers to see your score and an explanation for every question. Keep a calculator handy for questions 5, 6 and 8.
DiscussTalk about it
Try these with a classmate, a family member or in a notebook. There’s more than one good answer.
- Why is sin θ always less than 1 for an acute angle, while tan θ can be bigger than 1? What happens to tan θ as θ gets close to 90°? Try it in the explorer.
- Explain to someone who has never seen trig why every right triangle with a 35° angle has the same value of sin 35°, no matter how big it is.
- Think of something near you that you can’t measure directly: a flagpole, a building, the width of a river. What one distance and one angle would you measure to find it?
- Look at the exact-value table. Why is sin 30° equal to cos 60°? Can you find a rule for sin θ and cos (90° − θ)?
CreditsSources and further reading
- Jay Abramson et al., Algebra and Trigonometry 2e (OpenStax, Rice University), Section 7.2: Right Triangle Trigonometry and Section 8.3: Inverse Trigonometric Functions.
- Jay Abramson et al., Precalculus 2e (OpenStax, Rice University), Section 5.4: Right Triangle Trigonometry and Section 6.3: Inverse Trigonometric Functions.
- J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: Trigonometric functions, Hipparchus of Rhodes and Aryabhata I.
- Review the Pythagorean theorem used throughout this unit: Jugami Geometry: the Pythagorean Theorem.
Art credits: the surveyor scene, every triangle diagram, the explorer and the share image are original SVG drawn in code for Jugami. No outside images are used. All numbers on this page were checked by a script as well as by hand.