Right Triangle

Key Terms

Naming the Sides: Opposite, Adjacent, Hypotenuse

Defining Sine, Cosine, and Tangent

The Reciprocal Functions: Cosecant, Secant, Cotangent

Trigonometric Ratio Notation

Finding Missing Sides

Finding Missing Angles

The 45-45-90 Triangle

The 30-60-90 Triangle

Cofunction Relationships

Angles of Elevation and Depression

Limitations and Extensions

Exact Values at the Standard Acute Angles

Right Triangle Trigonometry FAQ

Where Trigonometry Begins: Ratios in a Right Triangle

The six trigonometric functions enter mathematics through a remarkably simple setup: a triangle with one right angle and two acute angles. Given any acute angle

\theta

in such a triangle, the three sides acquire names relative to that angle — opposite, adjacent, and hypotenuse — and the ratios between these sides define sine, cosine, tangent, and their reciprocals. What makes these ratios powerful is their invariance under scaling. Any two right triangles that share the same acute angle are similar, and similar triangles preserve all side ratios regardless of size. A 30-60-90 triangle with a hypotenuse of 2 and one with a hypotenuse of 200 yield identical values of

\sin(30°)

\cos(30°)

, and

\tan(30°)

. This scale-independence transforms a geometric observation into a universal computational tool — one that has been used for millennia to calculate distances that cannot be measured directly.

Right triangle trigonometry is limited to acute angles:

\theta

must satisfy

0° < \theta < 90°

. For angles outside this range — obtuse, negative, or beyond a full rotation — the definitions must be extended through the unit circle. But the exact values produced here, particularly from the special triangles, remain the foundation of all trigonometric computation. They populate the unit circle, appear in the graphs, drive the evaluation of trigonometric functions at standard angles, and recur in the solving of equations throughout the subject.

Key Terms

Triangle Sides

Hypotenuse— the side opposite the right angle, always the longest

Adjacent Side— the leg next to the chosen acute angle

Opposite Side— the leg across from the chosen acute angle

The Six Ratios

Sine— opposite over hypotenuse

Cosine— adjacent over hypotenuse

Tangent— opposite over adjacent

Cosecant— hypotenuse over opposite

Secant— hypotenuse over adjacent

Cotangent— adjacent over opposite

Supporting Concepts

Trigonometric Ratio— a ratio of sides depending only on the angle

Complementary Angles— the two acute angles sum to

90°

, producing cofunction relationships

Inverse Trigonometric Function— recovers an angle from a known side ratio

Formulas Used

Sine-Cosine Cofunction—

\sin\!\left(\frac{\pi}{2} - \theta\right) = \cos\theta

Cosine-Sine Cofunction—

\cos\!\left(\frac{\pi}{2} - \theta\right) = \sin\theta

Tangent-Cotangent Cofunction—

\tan\!\left(\frac{\pi}{2} - \theta\right) = \cot\theta

See All Trigonometry Definitions →

Naming the Sides: Opposite, Adjacent, Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle — always the longest side. The other two sides are called legs, and their names depend on which acute angle is under consideration. For a given acute angle

\theta

opposite side is the leg directly across the triangle from $\theta$ .
adjacent side is the leg that forms one ray of $\theta$ (the other ray being the hypotenuse).

If the triangle has acute angles

\alpha

and

\beta

, what is opposite for

\alpha

is adjacent for

\beta

, and vice versa. The hypotenuse remains the hypotenuse regardless of which angle is chosen — it is determined by the right angle, not by the acute angle under consideration.

This labeling is relational, not absolute. The same side can be "opposite" or "adjacent" depending on which angle is the point of reference. Keeping this distinction clear is essential, because the entire definition of the trigonometric functions depends on correctly identifying which side plays which role relative to the angle in question.

Defining Sine, Cosine, and Tangent

The three primary trigonometric functions are defined as ratios of the sides of a right triangle relative to an acute angle

\theta

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

The mnemonic SOH-CAH-TOA encodes these definitions: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Each ratio captures a different geometric relationship. Sine measures how much of the hypotenuse is "projected" onto the vertical direction relative to

\theta

. Cosine measures the horizontal projection. Tangent measures the steepness — how much vertical rise accompanies a given horizontal run. When

\theta

is small, the opposite side is short relative to the adjacent, and

\tan(\theta)

is small. As

\theta

approaches

90°

, the opposite side grows relative to the adjacent, and

\tan(\theta)

increases without bound.

Because the hypotenuse is always the longest side of a right triangle, both

\sin(\theta)

and

\cos(\theta)

are always between 0 and 1 for acute angles. Tangent has no such restriction — it can take any positive value, approaching infinity as

\theta

nears

90°

.

A fundamental observation: these ratios do not depend on the size of the triangle. If two right triangles share the same acute angle

\theta

, they are similar, and all corresponding side ratios are equal. A right triangle with legs 3 and 4 and hypotenuse 5 gives

\sin(\theta) = \frac{3}{5}

for the angle opposite the side of length 3 — and any similar triangle, no matter how large or small, produces the same value. The trigonometric functions are properties of the angle alone.

The function names themselves are standardized abbreviations — sin, cos, tan and their companions are catalogued among the trigonometry symbols together with the LaTeX code for each, and the mathematical keyboard lets you type them directly.

The Reciprocal Functions: Cosecant, Secant, Cotangent

Each primary ratio has a reciprocal:

\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{1}{\sin(\theta)}

\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{1}{\cos(\theta)}

\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan(\theta)}

These three functions — cosecant, secant, and cotangent — do not introduce new information. They repackage the same side relationships in inverted form. If

\sin(\theta) = \frac{3}{5}

, then

\csc(\theta) = \frac{5}{3}

. If

\cos(\theta) = \frac{4}{5}

, then

\sec(\theta) = \frac{5}{4}

.

In basic triangle-solving problems, the reciprocal functions appear less frequently than sine, cosine, and tangent. Their importance grows substantially in the study of identities, where expressions involving

\sec^2\theta

and

\csc^2\theta

arise from the Pythagorean identity and its variants. They also appear naturally in calculus — the derivatives and integrals of tangent and cotangent involve secant and cosecant — and in various formulas throughout physics and engineering.

Since the hypotenuse is always longer than either leg,

\csc(\theta) > 1

and

\sec(\theta) > 1

for all acute angles. Cotangent, like tangent, is unbounded:

\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}

can take any positive value.

The six ratios — three primary and three reciprocal — collect into one definitional reference card.

Function	Ratio of sides	Mnemonic / reciprocal of
sin θ	opposite / hypotenuse	SOH
cos θ	adjacent / hypotenuse	CAH
tan θ	opposite / adjacent	TOA
csc θ	hypotenuse / opposite	reciprocal of sin
sec θ	hypotenuse / adjacent	reciprocal of cos
cot θ	adjacent / opposite	reciprocal of tan

Trigonometric Ratio Notation

Notation

Trigonometric Ratio Notation

The subject's working marks are born on this page: three side labels that move with the angle, and six three-letter operator names. What they are not — products, cancellable factors — matters as much as what they are.

The angle letter

\theta

, the degree mark and radians are covered with degrees and radians; the inverse marks (

\sin^{-1}

, arcsin) belong to inverse functions.

\text{opp}

\text{adj}

\text{hyp}

opposite, adjacent, hypotenuse

Labels, not names: opp and adj are assigned relative to the chosen angle and swap the moment you switch to the other acute angle — Naming the Sides above. Only hyp is absolute, fixed by the right angle.

CasesDefinitions abbreviate freely —

\frac{\text{opp}}{\text{hyp}}

in the ratio formulas; once general triangle-solving starts, single letters take over under the triangle-labeling convention of side

a

facing angle

A

Do not confuseFixed properties of the drawing. Reading "adjacent" as one particular leg of the figure is the classic setup error — the labels must be reassigned every time the reference angle changes.

\sin

\cos

\tan

sine, cosine, tangent

Three-letter operator names, not products:

\sin\theta

applies a function to

\theta

the way

f(x)

does in function notation — nothing here multiplies. Print sets them upright (\sin in LaTeX); italic

sin

is a typesetting error.

CasesParentheses are optional while the argument is a single symbol —

\sin\theta

and

\sin(\theta)

say the same thing; they return the moment the argument compounds:

\sin(\theta + \phi)

, never

\sin\theta + \phi

Also written

\operatorname{sen}

— Spanish, Portuguese and Italian texts, from seno;

\operatorname{tg}

for tangent in Russian and Eastern European convention.

Do not confuseA factor named sin. "Cancelling"

\sin

from

\frac{\sin\alpha}{\sin\beta}

to get

\frac{\alpha}{\beta}

is the subject's signature error — an operator without its argument is not a quantity, and

\sin

alone means nothing.

\csc

\sec

\cot

cosecant, secant, cotangent

The reciprocal trio's abbreviations — The Reciprocal Functions above. The pairing is deliberately crosswise:

\csc

inverts

\sin

and

\sec

inverts

\cos

— the c-word goes with the s-word each time.

CasesSquares attach to the abbreviation itself — the

\sec^2\theta

and

\csc^2\theta

of the identities, shorthand for

(\sec\theta)^2

Also written

\operatorname{cosec}

— the British and Indian-curriculum spelling of cosecant;

\operatorname{ctg}

for cotangent in the same Eastern European texts that write

\operatorname{tg}

Do not confuseThe wrong partner. "Sec pairs with sine" reads naturally and is false — the reciprocal of

\sin

\csc

, never

\sec

; the crosswise rule catches everyone exactly once.

All trigonometry symbols →↑ Trigonometry

Finding Missing Sides

When one acute angle and one side of a right triangle are known, the remaining two sides can be found using the trigonometric ratios. The strategy is to choose the ratio that connects the known side, the unknown side, and the given angle.

Suppose a right triangle has an acute angle

\theta = 35°

and a hypotenuse of length 20. To find the opposite side, use sine:

\sin(35°) = \frac{x}{20} \quad \Rightarrow \quad x = 20\sin(35°)

To find the adjacent side, use cosine:

\cos(35°) = \frac{y}{20} \quad \Rightarrow \quad y = 20\cos(35°)

If instead the adjacent side is known (say, 15) and the opposite side is sought, tangent is the appropriate ratio:

\tan(35°) = \frac{x}{15} \quad \Rightarrow \quad x = 15\tan(35°)

The choice of ratio is not arbitrary — it is dictated by which pair of sides (one known, one unknown) relates to the given angle. Using the wrong ratio is one of the most common errors in triangle problems. A reliable method: label the sides as opposite, adjacent, or hypotenuse relative to the given angle, identify which two are involved (one known, one unknown), and select the ratio that pairs exactly those two.

Calculator usage requires attention to the angle mode. If the angle is given in degrees, the calculator must be in degree mode; if in radians, radian mode. A mismatch produces incorrect results with no error message — the calculator computes the wrong function value silently. This is a persistent source of mistakes, especially early in the subject.

Finding Missing Angles

When two sides of a right triangle are known, the acute angles can be determined. The process reverses the side-finding procedure: compute the ratio from the known sides, then apply the appropriate inverse trigonometric function to recover the angle.

For example, in a right triangle with opposite side 5 and hypotenuse 13:

\sin(\theta) = \frac{5}{13} \quad \Rightarrow \quad \theta = \arcsin\left(\frac{5}{13}\right) \approx 22.6°

Given adjacent side 7 and opposite side 10:

\tan(\theta) = \frac{10}{7} \quad \Rightarrow \quad \theta = \arctan\left(\frac{10}{7}\right) \approx 55.0°

Once one acute angle is found, the other follows immediately: the two acute angles in a right triangle always sum to

90°

. If

\theta \approx 22.6°

, the other acute angle is approximately

90° - 22.6° = 67.4°

.

Any pair of sides is sufficient. If the hypotenuse and adjacent side are known, use

\arccos

. If the opposite and adjacent sides are known, use

\arctan

. If the opposite and hypotenuse are known, use

\arcsin

. The Pythagorean theorem can also provide the third side first, opening up all three options — but this is an extra step that is rarely necessary.

The solver patterns for the most common right-triangle problems — six scenarios distinguished by what is given and what is sought — collect into the reference table below, covering both side-finding (sections above) and angle-finding cases.

Given	Find	Formula
angle θ + hypotenuse	opposite side	opposite = hypotenuse · sin θ
angle θ + hypotenuse	adjacent side	adjacent = hypotenuse · cos θ
angle θ + adjacent side	opposite side	opposite = adjacent · tan θ
opposite + hypotenuse	angle θ	θ = arcsin(opposite / hypotenuse)
adjacent + hypotenuse	angle θ	θ = arccos(adjacent / hypotenuse)
opposite + adjacent	angle θ	θ = arctan(opposite / adjacent)

The 45-45-90 Triangle

An isosceles right triangle — two equal legs and a right angle — has acute angles of

45°

each. Its side ratios are

1 : 1 : \sqrt{2}

, meaning that if each leg has length

a

, the hypotenuse has length

a\sqrt{2}

.

This follows from the Pythagorean theorem:

a^2 + a^2 = c^2

gives

c = a\sqrt{2}

.

Alternatively, the triangle arises by cutting a square along its diagonal. The diagonal of a unit square has length

\sqrt{2}

, and each resulting right triangle has legs of length 1 and hypotenuse

\sqrt{2}

.

The trigonometric values follow directly from the side ratios:

\sin(45°) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}, \quad \cos(45°) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}, \quad \tan(45°) = \frac{1}{1} = 1

The equality

\sin(45°) = \cos(45°)

is a direct consequence of the triangle's symmetry: the two legs are equal, so opposite and adjacent are identical. This is the only acute angle for which sine and cosine are equal. Tangent equals 1, meaning the opposite and adjacent sides have the same length — the angle bisects the range between

0°

and

90°

in a very concrete geometric sense.

The reciprocal values are

\csc(45°) = \sqrt{2}

\sec(45°) = \sqrt{2}

, and

\cot(45°) = 1

The 30-60-90 Triangle

The 30-60-90 triangle has side ratios

1 : \sqrt{3} : 2

, where the shortest side is opposite the

30°

angle, the medium side (of length

\sqrt{3}

relative to the shortest) is opposite the

60°

angle, and the hypotenuse is twice the shortest side.

This triangle arises from bisecting an equilateral triangle. An equilateral triangle with side length 2 has all angles equal to

60°

. Dropping an altitude from one vertex to the opposite side bisects that side (creating a segment of length 1) and bisects the vertex angle (creating a

30°

angle). The altitude has length

\sqrt{2^2 - 1^2} = \sqrt{3}

by the Pythagorean theorem. The result is a right triangle with angles

30°

60°

90°

and sides

1

\sqrt{3}

2

.

The trigonometric values at

30°

\sin(30°) = \frac{1}{2}, \quad \cos(30°) = \frac{\sqrt{3}}{2}, \quad \tan(30°) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

The trigonometric values at

60°

\sin(60°) = \frac{\sqrt{3}}{2}, \quad \cos(60°) = \frac{1}{2}, \quad \tan(60°) = \sqrt{3}

Notice the swap:

\sin(30°) = \cos(60°)

and

\cos(30°) = \sin(60°)

. This is the cofunction relationship at work —

30°

and

60°

are complements, so the sine of one equals the cosine of the other.

Together with the 45-45-90 values, these produce exact trigonometric values at

30°

45°

, and

60°

— the three non-trivial standard angles in the first quadrant. Combined with the values at

0°

(

\sin = 0

\cos = 1

) and

90°

(

\sin = 1

\cos = 0

), they form a complete set that extends to all quadrants via the unit circle.

Cofunction Relationships

In any right triangle, the two acute angles are complementary: they sum to

90°

(or

\frac{\pi}{2}

radians). This forces a structural relationship between the trigonometric functions evaluated at complementary angles.

Consider an acute angle

\theta

in a right triangle. The other acute angle is

90° - \theta

. The side that is opposite

\theta

is adjacent to

90° - \theta

, and the side that is adjacent to

\theta

is opposite

90° - \theta

. The hypotenuse is shared. Therefore:

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \cos(90° - \theta)

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \sin(90° - \theta)

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \cot(90° - \theta)

\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \csc(90° - \theta)

Each function, evaluated at an angle, equals its cofunction evaluated at the complementary angle. The prefix "co-" in cosine, cotangent, and cosecant is not accidental — it stands for "complement." Cosine is literally the sine of the complement. Cotangent is the tangent of the complement. Cosecant is the secant of the complement.

These relationships are verified by the special triangles:

\sin(30°) = \frac{1}{2} = \cos(60°)

\tan(45°) = 1 = \cot(45°)

(since

45°

is its own complement). They generalize beyond right triangles into the cofunction identities, which hold for all angles — not just acute ones — through the unit circle extension:

Sine-Cosine Cofunction

\sin\!\left(\frac{\pi}{2} - \theta\right) = \cos\theta

Learn more about this formula: Sine-Cosine Cofunction →

Browse all trigonometry formulas →

Cosine-Sine Cofunction

\cos\!\left(\frac{\pi}{2} - \theta\right) = \sin\theta

Learn more about this formula: Cosine-Sine Cofunction →

Browse all trigonometry formulas →

Tangent-Cotangent Cofunction

\tan\!\left(\frac{\pi}{2} - \theta\right) = \cot\theta

Learn more about this formula: Tangent-Cotangent Cofunction →

Browse all trigonometry formulas →

Secant and cosecant follow the same pattern:

\sec\!\left(\frac{\pi}{2} - \theta\right) = \csc\theta

.

The three function-cofunction pairings emerge directly from the complementary-angle structure of the right triangle, summarized below.

Function	Cofunction	Identity
sine	cosine	sin θ = cos(90° − θ)
tangent	cotangent	tan θ = cot(90° − θ)
secant	cosecant	sec θ = csc(90° − θ)

Angles of Elevation and Depression

Many practical problems involving right triangles are framed in terms of angles of elevation and depression. Both are measured from the horizontal — not from the vertical, and not from the ground.

An angle of elevation is the angle measured upward from a horizontal line of sight to a point above the observer. Standing on the ground and looking up at the top of a building, the angle between the horizontal and the line of sight to the top is the angle of elevation.

An angle of depression is the angle measured downward from a horizontal line of sight to a point below the observer. Standing on top of a cliff and looking down at a boat, the angle between the horizontal and the line of sight to the boat is the angle of depression.

A key geometric fact simplifies many problems: when a horizontal line from the observer and a horizontal line through the target are parallel (as they typically are, since both are horizontal), the angle of elevation from the lower point equals the angle of depression from the upper point. They are alternate interior angles formed by a transversal crossing two parallel lines. This means the problem can often be set up from either perspective.

The standard problem structure: the observer stands at a known distance from the base of an object, measures the angle of elevation to the top, and seeks the height. This produces a right triangle where the known distance is the adjacent side, the unknown height is the opposite side, and tangent is the relevant ratio:

\tan(\theta) = \frac{\text{height}}{\text{distance}} \quad \Rightarrow \quad \text{height} = \text{distance} \times \tan(\theta)

Variations include: finding the distance given the height and angle, finding the angle given the height and distance, problems involving two observation points at different distances, and problems where the observer's own height adds to the triangle.

Limitations and Extensions

Right triangle trigonometry is powerful but confined. It applies only to acute angles — those strictly between

0°

and

90°

. The definitions break down at the boundaries: at

0°

the "opposite" side has zero length and the triangle degenerates into a line segment, and at

90°

the "adjacent" side vanishes. Beyond

90°

, no right triangle can contain the angle as an acute angle, and the ratio definitions cease to apply.

This limitation motivates the transition to the unit circle, which redefines sine and cosine as coordinates of a point on a circle rather than as ratios in a triangle. The unit circle definition agrees with the right triangle definition for acute angles (the triangle formed by dropping a perpendicular from the circle to the

x

-axis reproduces the same ratios) and seamlessly extends to all real numbers.

For triangles that contain no right angle — oblique triangles — neither the right triangle ratios nor a direct application of the unit circle is sufficient. These require the Law of Sines and Law of Cosines, which generalize trigonometric relationships to arbitrary triangles. The Law of Cosines, in particular, reduces to the Pythagorean theorem when one angle is

90°

, revealing it as a direct extension of right triangle geometry.

Despite these limitations, the exact values and geometric intuitions established in right triangle trigonometry remain relevant throughout the entire subject. The special triangle values populate every quadrant of the unit circle, appear in every table of standard angles, and are used in the evaluation of trigonometric functions, the solving of equations, and the application of formulas. Right triangle trigonometry is where the subject begins, and its results persist everywhere the subject goes.

Exact Values at the Standard Acute Angles

The right triangle ratios and the two special triangles converge on a single deliverable: a table of exact values for all six trigonometric functions at the standard acute angles

0°

30°

45°

60°

, and

90°

. These values are the foundation of every quadrant of the unit circle, every standard-angle evaluation throughout trigonometry, and every exact computation in equations, identities, and applications.

Function	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	√3/3	1	√3	undefined
csc θ	undefined	2	√2	2√3/3	1
sec θ	1	2√3/3	√2	2	undefined
cot θ	undefined	√3	1	√3/3	0

Right Triangle Trigonometry FAQ

What is SOH CAH TOA?

A mnemonic for the three basic ratios. SOH gives sine as opposite over hypotenuse, CAH gives cosine as adjacent over hypotenuse, and TOA gives tangent as opposite over adjacent. Each ratio ties one acute angle of a right triangle to two of its sides, which is what makes missing sides and angles recoverable from partial information.Read more →

Which side is the opposite and which is the adjacent?

Both labels depend on the angle you are working from, not on the triangle. The hypotenuse is fixed, always facing the right angle. Of the remaining two sides, the opposite is the one across from your chosen angle and the adjacent is the one touching it. Switch to the other acute angle and the two swap places.Read more →

What are the special right triangles?

The 45-45-90 triangle has sides in the ratio 1 : 1 : √2, and the 30-60-90 triangle has sides in the ratio 1 : √3 : 2. Because the ratios are exact, they give exact trigonometric values at those angles without a calculator, and they are where the standard values on the unit circle come from.Read more →

Why does SOH CAH TOA only work for right triangles?

The three ratios are defined against the hypotenuse and the right angle, so a triangle without a right angle has neither. Removing the right angle removes the definitions themselves. Triangles of any shape need the law of sines and law of cosines instead, which reduce to the familiar ratios when one angle happens to be 90 degrees.Read more →

Why do sine and cosine of complementary angles swap?

Because the two acute angles share the same sides under opposite labels. What is opposite one angle is adjacent to the other, so the ratio defining sine for the first defines cosine for the second. That gives sin of an angle equal to cos of its complement, which is where the name cofunction comes from. Explore it in the angle explorer.Read more →