What is the sine of an angle?

In a right triangle, the sine of an acute angle is the ratio of the leg opposite that angle to the hypotenuse: sin α = a/c. In a 3-4-5 triangle where the side of length 3 lies opposite α, sin α = 3/5 = 0.6.

What is the difference between sine and cosine?

Both divide by the hypotenuse, but by different legs: the sine takes the leg opposite the angle (a/c), the cosine the leg next to it (b/c). That is why sin α = cos(90° − α) — the same side is "opposite" one acute angle and "adjacent" to the other.

How do you calculate the tangent?

The tangent is the ratio of the opposite leg to the adjacent leg: tan α = a/b. Equivalently tan α = sin α / cos α, because dividing a/c by b/c cancels the hypotenuse. English writes it "tan"; Polish textbooks use the abbreviation "tg" for the same function.

Why does the sine depend only on the angle and not on the size of the triangle?

Because two right triangles with the same acute angle are similar — one is a scaled copy of the other. Scaling multiplies every side by the same number, so their ratio does not change. The sine of 30° is one and the same number in a triangle 1 cm across and in one 1 km across.

What are sin 30°, sin 45° and sin 60°?

sin 30° = 1/2, sin 45° = √2/2 ≈ 0.71 and sin 60° = √3/2 ≈ 0.87. These are exact values: they follow from half an equilateral triangle and from the diagonal of a square, not from a measurement. Writing 0.87 gives only an approximation of one of them.

How do you find a side of a right triangle from an angle?

Rearrange the definition. Opposite leg: a = c · sin α. Adjacent leg: b = c · cos α. Hypotenuse from the opposite leg: c = a / sin α. For c = 12 cm and α = 35°, a = 12 · sin 35° ≈ 12 · 0.5736 ≈ 6.9 cm.

Why does my calculator give the wrong sine?

Almost always because it is in RAD or GRAD mode instead of DEG. In DEG mode sin 30 = 0.5; in RAD mode the same keystrokes give about −0.988, because the calculator reads 30 as radians. Check for DEG on the display before you compute anything.

Can a sine be greater than 1?

Not in a right triangle. Sine and cosine divide a leg by the hypotenuse, and the hypotenuse is always the longest side, so the result stays between 0 and 1. The tangent has no such bound — near 90° it grows without limit.

Intermediate level

Sine, cosine and tangent

Sine, cosine and tangent are three ratios of the sides of a right triangle. See why they depend on the angle alone, where the exact values for 30°, 45° and 60° come from, and how to find a side you cannot measure.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

The Pythagorean theoremIn a right triangle a² + b² = c². See why it works, how to find the hypotenuse and a leg, what Pythagorean triples are, and how the converse lets you check whether a corner really is square.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Wheelchair ramps
The step up to the terrace is 45 cm and a wheelchair needs a ramp no steeper than 6%. A gradient is the tangent of the angle, so tan α = 0.06, meaning α ≈ 3.4°, and the horizontal run comes to 45 cm / 0.06 = 750 cm. The ramp has to be 7.5 m long — usually more than the space in front of the terrace, which is why such ramps end up as two shorter flights with a landing.
Measuring heights
No tape reaches the top of the tree over your house, but any phone with a spirit level reads an angle. Step 15 m back from the trunk, sight the crown and the phone says 52°. The height is 15 · tan 52° = 19.2 m plus the 1.6 m your eyes are off the ground, so 20.8 m in all. The roof is 12 m away, so the tree would clear it — and that is the number the tree surgeon wants.
Solar panels
A row of panels 1.7 m across, tilted at 30°, rises 1.7 · sin 30° = 0.85 m above the roof. At noon in December the sun in Poland sits about 17° above the horizon, so the shadow runs 0.85 / tan 17° = 2.8 m. That is the gap the installer leaves before the next row; anything tighter and the front row shades the one behind it for half the day, in the months with least light to spare.

All formulas

Sine
$\sin \alpha = \frac{a}{c}$
the leg opposite the angle over the hypotenuse
Cosine
$\cos \alpha = \frac{b}{c}$
the leg adjacent to the angle over the hypotenuse
Tangent
$\tan \alpha = \frac{a}{b} = \frac{\sin \alpha}{\cos \alpha}$
the opposite leg over the adjacent leg
The opposite leg
$a = c \cdot \sin \alpha$
the definition of the sine solved for a
The adjacent leg
$b = c \cdot \cos \alpha$
the definition of the cosine solved for b
The hypotenuse
$c = \frac{a}{\sin \alpha}$
when the leg opposite the angle is known
Special angles
$\sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}$
exact values, not readings from a table

The Pythagorean theorem relates the three sides of a right triangle. Trigonometry adds a fourth quantity — the angle — and lets you compute a side from just one other side and one angle.

Opposite and adjacent

Before any formula appears, the sides have to be named relative to a chosen acute angle. This is the one place where readers go wrong more often than in the arithmetic.

With respect to α: side a lies opposite it, side b next to it, and the hypotenuse c opposite the right angle.

The hypotenuse $c$ — the side opposite the right angle, always the longest. It does not depend on which acute angle you pick.
The opposite leg $a$ — it does not touch the angle $\alpha$ .
The adjacent leg $b$ — it forms the angle $\alpha$ together with the hypotenuse.

Pick the other acute angle and $a$ and $b$ swap roles. The hypotenuse stays where it was.

The three ratios

\sin \alpha = \frac{a}{c}, \qquad \cos \alpha = \frac{b}{c}, \qquad \tan \alpha = \frac{a}{b}

In words: the sine divides the opposite side by the hypotenuse, the cosine divides the adjacent side by the hypotenuse, and the tangent divides the opposite side by the adjacent one.

The tangent is not a separate idea — it is the quotient of the other two:

\frac{\sin \alpha}{\cos \alpha} = \frac{a/c}{b/c} = \frac{a}{b} = \tan \alpha

In a right triangle with a = 3, b = 4, c = 5, find the sine, cosine and tangent of the angle α opposite side a.

Why only the angle matters

Two right triangles with the same acute angle are similar — one is simply a scaled copy of the other. Scaling multiplies every side by the same number $k$ :

\frac{k \cdot a}{k \cdot c} = \frac{a}{c}

The scale cancels inside the fraction. That is why $\sin 30^\circ$ is one number — the same in a triangle with a hypotenuse of $1\,\text{cm}$ and in one with a hypotenuse of $1\,\text{km}$ . It is the entire reason tables of values can exist at all.

Special angles: exact values

For three angles the values can be derived with no table and no calculator — and they come out exact, with a square root instead of a decimal expansion.

45° — take a square of side $1$ and cut it along the diagonal. You get a right triangle with legs $1$ and $1$ , and its hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$ . Hence

\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

30° and 60° — take an equilateral triangle of side $2$ and cut it along its height. You get a right triangle with hypotenuse $2$ , shorter leg $1$ (half the side) and height $\sqrt{2^2 - 1^2} = \sqrt{3}$ . Opposite the $30^\circ$ angle lies the side $1$ , opposite the $60^\circ$ angle the side $\sqrt{3}$ :

\sin 30^\circ = \frac{1}{2}, \qquad \sin 60^\circ = \frac{\sqrt{3}}{2}

$\alpha$	$\sin \alpha$	$\cos \alpha$	$\tan \alpha$
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	$1$	$0$	undefined

Two things are worth spotting in this table. First, the cosine column is the sine column read bottom-up — because $\cos \alpha = \sin(90^\circ - \alpha)$ , which is the same side viewed from the other acute angle. Second, $\tan 90^\circ$ is undefined: at $90^\circ$ the adjacent side would have length zero, and division by zero is not a thing.

$\frac{\sqrt{3}}{2}$ is the exact value; $0.87$ is its two-decimal approximation. When a question asks for the exact value, type the form with the root.

Finding a side

Each definition can be solved for the length you are after:

a = c \cdot \sin \alpha, \qquad b = c \cdot \cos \alpha, \qquad c = \frac{a}{\sin \alpha}

The recipe never changes: name the sides relative to the given angle, pick the function that links the known side to the unknown one, and only then substitute numbers.

A right triangle has a hypotenuse of 12 cm and an acute angle of 35° at it. Find the leg opposite that angle (to 0.1 cm).

A ladder leans against a wall at 70° to the ground, with its foot 1.2 m from the wall. How long is the ladder (to 0.1 m)?

The calculator: DEG mode

A calculator computes the sine in whatever angle measure it is set to, and the abbreviation on the display tells you which:

DEG — degrees. $\sin 30 = 0.5$ . This is the mode school problems are solved in.
RAD — radians. The same keystrokes give $\sin 30 \approx -0.988$ , because $30$ is read as $30$ radians.
GRAD — gradians (a full turn is $400^\text{g}$ ), used in surveying.

Before computing anything, press $\sin 30$ and check that you see $0.5$ . If you do not, switch the mode — not the answer.

Exercises

For a special angle, type the exact value, e.g. √3/2 or 1/2 (a two-decimal approximation is accepted as well — a one-decimal one is not). For a side length the unit is given in brackets, so type the bare number rounded to 0.1.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

cos 45° =

Common mistakes

Swapping the opposite and the adjacent leg — name them before you reach for a formula; this causes most of the errors.
Using the sine in a triangle with no right angle — these three definitions only hold in a right triangle. For any triangle, use the laws of sines and cosines.
A calculator left in RAD mode — the most common source of an answer that is wildly off.
Quoting $0.87$ as the exact value — the exact value is $\frac{\sqrt{3}}{2}$ .
Multiplying instead of dividing — when the unknown is in the denominator ( $\cos \alpha = \frac{b}{c}$ , solving for $c$ ), you have to divide.
Treating $\tan 90^\circ$ as a number — that tangent does not exist.

Formula card

Topic: Sine, cosine and tangent

Sine
$\sin \alpha = \frac{a}{c}$
the leg opposite the angle over the hypotenuse
Cosine
$\cos \alpha = \frac{b}{c}$
the leg adjacent to the angle over the hypotenuse
Tangent
$\tan \alpha = \frac{a}{b} = \frac{\sin \alpha}{\cos \alpha}$
the opposite leg over the adjacent leg
The opposite leg
$a = c \cdot \sin \alpha$
the definition of the sine solved for a
The adjacent leg
$b = c \cdot \cos \alpha$
the definition of the cosine solved for b
The hypotenuse
$c = \frac{a}{\sin \alpha}$
when the leg opposite the angle is known
Special angles
$\sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}$
exact values, not readings from a table

A right triangle seen from the angle α: side a lies opposite it, side b next to it, and the hypotenuse c opposite the right angle.

Frequently asked questions

A luminous unit circle with a radius sweeping out an angle, a sine wave unrolling from it as a streak of light, beside a right triangle drawn in stars, against a cosmic nebula in violet and magenta

Trigonometry

Why sine doesn't depend on the size of the triangle

The height of a tower, the width of a river, the distance to a ship offshore — all of it can be computed without going anywhere near the thing being measured. The whole of trigonometry rests on one observation: the ratio of two sides in a right triangle depends only on the angle, not on how big the triangle is. That is what turns an angle into a number — and lets it be tabulated once and for all.

13 min