Trigonometric functions

You first name the sides of a right triangle and turn their lengths into six functions of an acute angle. Identities connect those functions and help you recover missing values, rebuild special-angle values and verify equations. Signed coordinates extend the same definitions to every angle, including axes and all four quadrants. The unit circle turns any real number into an angle input, and matching calculator mode gives decimal approximations.

Lessons

Vocabulary

Right triangle ryt TRY-ang-gul: A triangle with one 90° corner. The other two angles are acute and add up to 90°. Like: A ladder leaning against a wall, with the floor.
Acute angle uh-KYOOT ANG-gul: An angle greater than 0° and less than 90°, narrower than a square corner. Like: A pizza slice thinner than a quarter of the pizza.
Hypotenuse (hyp) hy-POT-uh-noos: The side across from the right angle. It is always the longest side of the right triangle. Like: The ladder itself, reaching from the floor to the wall.
Opposite side (opp) OP-uh-zit syd: The leg across the triangle from the angle θ you are working with. It does not touch θ. Like: Standing at the ladder's foot, the wall across from you.
Adjacent side (adj) uh-JAY-sunt syd: The leg that touches the angle θ and is not the hypotenuse. Like: The strip of floor between the ladder's foot and the wall.
With respect to (w.r.t.) with ree-SPEKT too: From the point of view of. Opposite and adjacent are always named with respect to one chosen angle. Like: Left and right depend on which way you face.
Pythagorean theorem puh-THAG-uh-REE-un THEER-um: In a right triangle with legs a and b and hypotenuse c, $a^{2}$ + $b^{2}$ = $c^{2}$ . It finds a missing side. Like: Two small square tiles together match the area of one big tile.
Similar triangles SIM-uh-lur TRY-ang-gulz: Triangles with the same angles. One is a scaled copy of the other, so the fractions made from their sides are equal. Like: A photo and its enlargement.
Trigonometric function trig-uh-nuh-MET-rik FUNK-shun: A rule that takes an angle and returns a ratio of two sides, or of coordinates. There are six: sin, cos, tan, cot, sec, csc. Like: A vending machine: put in an angle, get out a number.
Sine (sin θ) syn: $\frac{o p p}{h y p}$ in a right triangle, $\frac{y}{r}$ for a point on the terminal side, and the y-coordinate on the unit circle. Like: The height a ladder reaches, per foot of ladder.
Cosine (cos θ) KOH-syn: $\frac{a d j}{h y p}$ in a right triangle, $\frac{x}{r}$ for a point on the terminal side, and the x-coordinate on the unit circle. Like: How far the ladder's foot sits from the wall, per foot of ladder.
Tangent (tan θ) TAN-junt: For acute angles, opposite over adjacent. Generally $\frac{y}{x}$ or $\frac{\sin θ}{\cos θ}$ , defined when x ≠ 0. Like: The steepness of a ramp: rise over run.
Cotangent (cot θ) koh-TAN-junt: For an acute angle, the flip of tangent. Generally cot θ = $\frac{x}{y}$ or $\frac{\cos θ}{\sin θ}$ , defined when y ≠ 0. Use $1 over tan θ$ only when tangent is defined and nonzero. Like: Run over rise: a ramp's steepness turned upside down.
Secant (sec θ) SEE-kant: The reciprocal of nonzero cosine: $\frac{h y p}{a d j}$ for an acute triangle angle and $\frac{r}{x}$ for a point, defined when x ≠ 0. Like: Ladder length per foot of floor it covers.
Cosecant (csc θ) koh-SEE-kant: The reciprocal of nonzero sine: $\frac{h y p}{o p p}$ for an acute triangle angle and $\frac{r}{y}$ for a point, defined when y ≠ 0. Like: Ladder length per foot of height it reaches.
Reciprocal rih-SIP-ruh-kul: The flip of a nonzero number: 1 divided by it. A number times its reciprocal is 1. Zero has no reciprocal. Like: Turning a fraction upside down.
Identity eye-DEN-tuh-tee: An equation that is true for every value of the variable where both sides are defined. Like: Two different packages with exactly the same contents.
Reciprocal identities rih-SIP-ruh-kul eye-DEN-tuh-teez: csc θ = $1 over sin θ$ , sec θ = $1 over cos θ$ , cot θ = $1 over tan θ$ where both sides exist. The cotangent formula additionally needs a defined, nonzero tangent. Like: Three pairs of upside-down twins.
Quotient identities KWOH-shunt eye-DEN-tuh-teez: Express tangent as sine divided by nonzero cosine; express cotangent as cosine divided by nonzero sine. In acute triangles, cancelling the common hypotenuse produces opposite over adjacent for tangent. Like: Two fractions with the same bottom: divide them and the bottoms cancel.
Pythagorean identity puh-THAG-uh-REE-un eye-DEN-tuh-tee: $\sin^{2}$ θ + $\cos^{2}$ θ = 1. Dividing by nonzero $\cos^{2}$ θ gives $\tan^{2}$ θ + 1 = $\sec^{2}$ θ; dividing by nonzero $\sin^{2}$ θ gives $\cot^{2}$ θ + 1 = $\csc^{2}$ θ. Like: The Pythagorean theorem with the hypotenuse shrunk to 1.
$\sin^{2}$ θ syn skwaird THAY-tuh: Shorthand for (sin θ $) squared$ : find the sine, then square it. It does not mean sin( $θ^{2}$ ). Like: Squaring your test score: get the score first, then square it.
LHS el aych ess: Abbreviation for left side: the expression to the left of the equals sign. Like: The two pans of a balance scale.
Prove (verify) an identity proov or VAIR-uh-fy an eye-DEN-tuh-tee: Show that two expressions agree for every allowed angle by justified algebra. Transforming one side into the other is the beginner strategy used here. Like: Reducing $6 over 10$ step by step until it visibly becomes $3 over 5$ .
Special angles SPESH-ul ANG-gulz: 30°, 45° and 60° ( $π over 6$ , $π over 4$ , $π over 3$ ). Their trig values are exact fractions and roots, read from two triangles. Like: The few phone numbers you know by heart.
Exact value ig-ZAKT VAL-yoo: An answer with no rounding. Integers, fractions and roots can be exact; terminating decimals such as 0.5 can also be exact. Like: Saying one third instead of 0.33.
Rationalize the denominator RASH-uh-nuh-lyz thuh dih-NOM-uh-nay-tur: Rewrite a fraction so no square root is on the bottom. For the single-root denominators used here, multiply top and bottom by that nonzero root; a sum on the bottom requires a different method. Like: Repackaging the same contents in a standard box.
Standard position STAN-durd puh-ZISH-un: An angle with its vertex at the origin and its initial side along the positive x-axis. Like: Every runner starts at the same line, facing the same way.
Terminal side TUR-muh-nul syd: The ray where an angle ends after its rotation. The point definition reads a point on it. Like: Where a clock hand stops.
Rectangular coordinate system rek-TANG-gyuh-lur koh-OR-duh-nut SIS-tum: Two number lines, the x-axis and the y-axis, crossing at right angles at the origin (0, 0). Every point has an address (x, y). Like: A city map: blocks east, then blocks north.
Quadrant KWOD-runt: One of the four regions the axes cut the plane into, numbered I, II, III, IV counterclockwise from the upper right. Like: Four slices of a pizza cut with two straight perpendicular cuts.
r (distance to the origin) ar: The distance from the origin to P(x, y): r = $square root of x squared + y squared$ . It is positive when P is not the origin; the origin has distance zero. Like: The straight-line distance a bird flies home.
Quadrantal angle kwod-RAN-tul ANG-gul: An angle whose terminal side lies on an axis: 0°, 90°, 180°, 270°, 360°, and angles coterminal with them. Like: Clock hands pointing at 3, 12, 9 or 6.
Undefined un-di-FYND: No value exists, because the formula would divide by zero. Like: A vending machine with no button for your choice.
Domain DOH-mayn: All the inputs a function accepts. sin and cos accept every angle; tan and sec skip angles on the y-axis; cot and csc skip angles on the x-axis. Like: The list of coins a vending machine accepts.
Unit circle YOON-it SUR-kul: A circle of radius 1 centered at the origin, with equation $x^{2}$ + $y^{2}$ = 1. A general origin-centered circle has equation $x^{2}$ + $y^{2}$ = $r^{2}$ . Like: A round track whose every point is one step from the center pole.
Central angle SEN-trul ANG-gul: A central angle has its vertex at the circle’s center. For nonnegative radian measure θ, its subtended arc length s satisfies θ = $\frac{s}{r}$ . Signed travel uses direction separately. Like: A pizza slice: the tip is the center, the crust is the arc.
Radian RAY-dee-un: An angle unit: one radian cuts off an arc exactly as long as the radius. π radians = 180°. Like: Measuring a turn by the distance walked around a circle of radius 1.
Real number t on the unit circle REE-ul NUM-bur tee: Signed travel t from (1, 0): distance |t| around the unit circle, counterclockwise if t > 0 and clockwise if t < 0. It equals the signed radian angle. Like: Steps walked around a round track, forward or backward.
Terminal point P(t) TUR-muh-nul point pee of tee: The unit-circle point reached after traveling distance |t| from (1, 0), with direction given by the sign of t. Its coordinates are (cos t, sin t). Like: The spot on the track where you stop walking.
Circular functions SUR-kyuh-lur FUNK-shunz: Another name for the trig functions when they are defined on the unit circle and their inputs are real numbers. Like: A nickname that says where the functions come from.
DEG and RAD mode dee ee jee and ar ay dee mohd: The calculator setting that says whether the number you type is in degrees or radians. It must match the problem. Like: A kitchen scale set to grams or ounces.
RHS ar aych ess: Abbreviation for right side: the expression to the right of the equals sign. Like: The right pan of a scale.
Origin OR-uh-jin: The coordinate address (0, 0), where the two axes cross. Like: The starting point on a map.
Vertex VUR-teks: The meeting point of an angle’s two sides. Like: The hinge of a turning door.
Initial side ih-NISH-ul syd: The ray from which an angle’s rotation begins. Like: The position of a clock hand before it moves.
Coterminal angles koh-TUR-muh-nul ANG-gulz: Angles that end on the same ray after rotations differing by whole turns. Like: Stopping at the same track spot after an extra lap.
Radius RAY-dee-us: The distance from a circle’s center to any point on the circle. Like: A spoke of a wheel.
Arc ark: A curved part of a circle between two points. Like: The crust along a pizza slice.
Arc length s ark length ess: The nonnegative distance measured along an arc. For a signed turn θ, the traveled length is r|θ|, with θ in radians. Like: Distance walked along a curved track.
Absolute value AB-suh-loot VAL-yoo: A number’s distance from zero, written with bars. It is never negative. Like: How far you are from home, whichever direction you walk.
Angle θ ANG-gul THAY-tuh: θ is a Greek letter used as the name of an angle. α, β, γ and δ are other angle names. Like: Different name labels on the same kind of drawer.
Function FUNK-shun: A rule giving exactly one output for each allowed input. Trigonometric functions take angle inputs and produce numerical ratios. Like: A machine with one result for each accepted setting.
Range raynj: The set of outputs a function can produce. Sine and cosine have range from −1 to 1, including the endpoints. Like: The list of items a machine can deliver.
Difference of squares DIF-ur-ens uv skwairz: The product (a + b)(a − b) equals $a^{2}$ − $b^{2}$ because its middle products cancel. Like: Two opposing adjustments that undo one another.
Perfect square PUR-fikt skwair: A nonnegative whole number obtained by squaring a whole number. Its square root is a whole number. Like: The area of a square tile with whole-number side length.
Factor FAK-tur: A number or expression multiplied by another to make a product. A common factor multiplies the whole numerator and denominator. Like: One ingredient multiplied into a batch.
Prime prym: A positive integer greater than 1 whose only positive whole-number factors are 1 and itself. Like: A quantity that cannot form a smaller equal rectangular grouping.
Ratio RAY-shee-oh: A comparison of one quantity with another by division. The order says which quantity is on top. Like: Wall height per foot of ladder.
Leg leg: Either of the two sides meeting at the right angle of a right triangle. Each is shorter than the hypotenuse. Like: The floor or wall beside a leaning ladder.
Perpendicular pur-pun-DIK-yuh-lur: Meeting at a right angle, exactly 90°. Like: The square corner where a wall meets a level floor.
Circumference sur-KUM-fur-ens: The full distance around a circle, equal to 2πr for radius r. Like: The total length of one lap around a round track.

Quick checks

sin θ > 0 and cos θ < 0. Which quadrant is θ in?

Quadrant II: sin θ > 0 means y > 0 (up), and cos θ < 0 means x < 0 (left).

Find sec 420° and csc 405° exactly.

sec 420° = 2, because removing 360° gives cosine $1 over 2$ .
csc 405° = $square root of 2$ , because removing 360° gives sine $square root of 2 over 2$ , whose reciprocal is $square root of 2$ .

Find tan and cot .

tan $5 π over 2$ is undefined, because removing 2π gives the point (0, 1) and $\frac{y}{x}$ divides by zero.
cot $5 π over 2$ = 0, because $\frac{x}{y}$ = $0 over 1$ .

Can sin θ equal 1.2?

No. θ + θ = 1 and θ ≥ 0, so θ ≤ 1 and |sin θ| ≤ 1.

Rationalize .

= : multiply top and bottom by , then reduce to .

θ is acute and cos θ = . Find sin θ and tan θ.

sin θ = $12 over 13$ , because 1 − $25 over 169$ = $144 over 169$ and the acute-angle root is positive.
tan θ = $12 over 5$ , because $12 over 13$ ÷ $5 over 13$ = $12 over 5$ .

Is the point (, ) on the unit circle?

No: ( + ( = + = , not 1.

Why does sin 30 show −0.98803 in RAD mode?

With no degree sign, 30 means 30 radians, about 1718.87°, a different angle from 30°.

Before you start

Explain it like I am five
Picture a ladder leaning against a wall. You see three lengths: the height up the wall, the distance along the floor and the ladder itself. A steeper ladder reaches a larger share of its length up the wall.
Now replace it with a ladder twice as long at the same tilt. The height and floor distance both double too. The shares stay the same because both numbers in each fraction double.
The six trigonometric functions name six ways to compare those lengths. Later you turn the ladder around a center point. Map addresses with positive or negative directions keep track of left, right, up and down, so the same comparisons can describe any turn.
Read the symbols before doing arithmetic
The letters on a math page are labels, like names on drawers. θ, pronounced THAY-tuh, often labels an angle. α is AL-fuh, β is BAY-tuh, γ is GAM-uh, and δ is DEL-tuh. The angle letter changes the name, not the method. The symbol ≠ means is not equal to. The symbols < and > point their wide opening at the larger number. ≤ means less than or equal to; ≥ means greater than or equal to. The symbol ∞ means unbounded size, not a last number. The sign ° means degrees; a full turn is 360°. Ordinary letters such as x and u name numbers that may vary. Counterclockwise means opposite the way a clock hand turns. The symbol ≥ means the same as >=, and ≤ means the same as <=. A factor is a number multiplied by another number to make a product. In 3 × 4 = 12, both 3 and 4 are factors. The symbol × means multiply, ÷ means divide, and ² means multiply a number by itself. When single letters stand for numbers, touching letters mean multiplication: uw = u × w. A number before a letter also means multiplication: 2u = 2 × u. The symbol ± means plus or minus: ±2 gives the two choices 2 and −2. A nonzero number is any number except zero. Division by zero has no value: 4 ÷ 0 would require a number whose product with 0 is 4, but every such product is 0.
Fractions: what they mean and how to reduce them
A fraction is a division waiting to happen: $6 over 10$ means 6 ÷ 10. The bottom number says how many equal pieces the whole is cut into; the top says how many pieces you have. Six slices of a pizza cut into 10 is the same amount of pizza as 3 slices of a pizza cut into 5. Reducing a fraction means rewriting it with the smallest possible numbers without changing its size. Trig answers are always given reduced.
Squaring fractions and negatives, and subtracting from 1
Squaring means multiplying a number by itself: $3 squared$ = 3 × 3 = 9. To square a fraction, square the top and square the bottom. A negative number squared is positive, because a negative times a negative is positive. This section keeps computing things like 1 − ( $1 over 2$ $) squared$ , so you need three moves: square a fraction, rewrite 1 as a fraction, and subtract.
Square roots from zero
A square root undoes squaring. $square root of 9$ asks: which nonnegative number times itself makes 9? The answer is 3, because 3 × 3 = 9. The name comes from square tiles: a square tile with area 9 has sides of length $square root of 9$ = 3. Numbers such as 1, 4, 9, 16, 25 have whole-number roots and are called perfect squares. Most numbers do not: $square root of 3$ ≈ 1.732, rounded to three decimal places. Its full decimal never ends, so in an exact answer you leave it written as $square root of 3$ .
Simplifying a square root
$square root of 50$ is correct but not finished. The rule is to pull every perfect square out from under the root, so the number left inside is as small as possible. $square root of 50$ becomes 5 $square root of 2$ , read five root two, which means 5 × $square root of 2$ . Answer keys use this form, and you need it to recognize that your answer matches theirs. A number beside a root means multiplication: 2 $square root of 5$ means 2 × $square root of 5$ .
Rationalizing a denominator
By convention a finished answer has no square root on the bottom of a fraction. $1 over square root of 2$ and $square root of 2 over 2$ are the same number, about 0.707, but only the second is in finished form. In these single-root examples, rationalizing rewrites the fraction so its bottom is an integer, without changing its value. The tool is multiplying by a fraction that equals 1.
Dividing fractions: flip the second and multiply
Dividing asks how many times one amount fits into another. 6 ÷ 2 = 3 because three 2s fit into 6. With fractions, 3 ÷ $1 over 2$ asks how many half pizzas fit into 3 pizzas: 6. Dividing by $1 over 2$ gave the same result as multiplying by 2, and 2 is $1 over 2$ upside down. The flip is called the reciprocal of a nonzero number: 1 divided by that number. Multiplying a number by its reciprocal gives 1. That is the whole rule: to divide by a fraction, multiply by its flip. You will use this when two side ratios are divided to form a third ratio. Sign reminder: a negative divided by a positive is negative, and a negative divided by a negative is positive. For example $−6 over 3$ = −2 because (−2) × 3 = −6, while $−6 over −3$ = 2 because 2 × (−3) = −6.
The Pythagorean theorem: finding a missing side
A right triangle has a 90° corner. Its two sides meeting there are legs, a and b. The side across from that corner is the hypotenuse, c. Square the legs and add: their total is the square of the hypotenuse, $a^{2}$ + $b^{2}$ = $c^{2}$ . If you know two sides, this finds the third. Picture a square tile on each side: the two smaller areas together match the big area on the hypotenuse.
Absolute value means size without direction
Absolute value is distance from zero on a number line. Distance does not keep a minus sign. The bars in |−3| ask for the size of −3, which is 3. Think of walking three blocks left or right from home: either trip puts you three blocks away. Thus |3| = 3 too, and |0| = 0. You will use these bars to say that a function value stays within a certain size even when it is negative. Magnitude is another word for size: the magnitude of −3 is |−3| = 3.
Keep an equation balanced while isolating an unknown
An equation says two amounts are equal, like a balanced scale. You may subtract the same amount from both pans and keep the balance. Solving means finding every number that makes the equation true. Isolating an unknown means leaving it alone on one side so you can read or recover it. You will often isolate a squared number first, then undo the square.
Fractions with letters, brackets and common factors
Letters can stand for whole numbers or expressions. A fraction with letters obeys the same arithmetic as a fraction with numbers, like a box that can hold different objects without changing how its lid works. You may combine fractions after giving them the same bottom. You may distribute multiplication to each term inside brackets. You may cancel a factor that multiplies the entire top and bottom, but a piece added to another piece is not a shared factor. With single-letter number names, uw means u × w; 2u means 2 × u.
Points, quadrants and the distance to the origin
A point (x, y) is an address on a rectangular coordinate system: two number lines called axes that meet at 90°, meaning they are perpendicular. Their crossing is the origin, (0, 0). Go x steps right, left if x is negative, then y steps up, down if y is negative. The horizontal line is the x-axis, and the vertical line is the y-axis. They cut the plane into four quadrants, numbered I, II, III, IV counterclockwise from the upper right. Walking across and up forms two right-triangle legs; the Pythagorean theorem gives the straight distance home.
Degrees and radians for the special angles
Angles come in two units. Degrees cut a full turn into 360 equal parts. A diameter is the straight distance across a circle through its center. The fixed constant π, read pi, is about 3.14159: circumference divided by diameter. A radius is the straight distance from the center to the rim, half a diameter. The circumference is the total distance around the rim, 2πr for radius r. An arc is a curved portion of that rim. Radians measure arc length divided by radius. On a circle of radius 1, that is the distance traveled along the rim, so a full turn is 2π radians, the whole circumference, and a half turn is π radians. The one fact to hold onto: π radians = 180°. With it, $π over 6$ radians is 180° ÷ 6 = 30°.
Angle starts, signed turns and coterminal angles
An angle records a turn, like a door rotating on its hinge. The hinge is the vertex. A ray starts at a point and extends straight forever in one direction. The starting ray is the initial side, and the stopped ray is the terminal side. In standard position the hinge is at the origin and the initial side points right along the positive x-axis. Positive turns go counterclockwise; negative turns go clockwise. This is a convention chosen so everyone records the same turn with the same sign. A full turn brings the ray back to its starting direction.
Use a line equation to choose a point
A line equation is an address rule, like a street whose houses all obey the same map pattern. In y = −3x, once you choose the horizontal coordinate x, the rule determines the vertical coordinate y. A line through the origin has two opposite halves. A terminal side is only one ray, so a quadrant condition tells you which half to use. In Quadrant IV choose a positive x, which makes y negative in this rule.
Round a decimal at the requested place
Rounding chooses a nearby decimal with a fixed number of digits, like recording a measured length to the nearest marked notch on a ruler. Four decimal places means four digits after the decimal point. Keep the first four, then inspect the fifth. A fifth digit from 0 through 4 leaves the fourth unchanged; 5 through 9 increases the kept amount by one in its last place. Keep trailing zeros because they show the requested precision.