Intuition The ONE idea of this whole topicA point walking around a circle of radius 1 1 shadow on the floor (how far right it is) and its height (how far up it is) are the two legs of a right triangle whose slanted long side is that radius. Because "leg-square plus leg-square equals slant-square", those two distances are locked together forever — and that lock is the whole topic. Every symbol in the parent note is just a name for one of the pieces in this picture.
This page assumes nothing . Before you can even read the parent note's headline formula you must already own about a dozen tiny ideas. We build every one from zero, in the order they depend on each other — no symbol is used before its own section defines it . If a symbol shows up on the parent note and you can't picture it, it lives here.
We start here because everything else — directions, circles, distances — is drawn on this grid.
( x , y ) origin . The horizontal line is the ==x ; the vertical line is the y x right " of the origin (negative = left); the second number y up " (negative = down). A dot is pinned down by its pair ( x , y )
We need this because a "direction" will eventually land on an actual point , and we describe points with ( x , y )
The plane splits into four quadrants , and their sign patterns matter enormously later (they decide ±
Quadrant
where
x y
I
top-right
+ +
II
top-left
− +
III
bottom-left
− −
IV
bottom-right
+ −
The unit circle is defined by "distance 1 what distance means on the grid.
Definition Distance in the planeThe straight-line distance from the origin ( 0 , 0 ) ( x , y ) x y distance = x × x + y × y . = 1
Intuition Why we need a distance rule at all"The unit circle is all points at distance 1 always the slant of a right triangle — that is the only ruler we use in this whole chapter.
θ θ name for a direction , measured as a turn away from pointing along the positive x positive θ anticlockwise ; a negative θ clockwise (the opposite way).
Definition Units: degrees vs radiansA full loop is either 360 degrees (written 36 0 ∘ 2 π radians (no little circle). Both name the same turn. On this page we use degrees and always write the ∘ π /4
Look at the figure below. The amber ray starts flat along the right and swings up. The wedge between "flat right" and the ray is θ
Intuition Why we need an angle at allEvery quantity in this topic is a machine that eats an angle and spits out a number . Without a clear picture of what an angle is (a turn, not a shape), none of the machines mean anything.
θ = 0 ∘ θ = 9 0 ∘ θ = 18 0 ∘ θ = − 9 0 ∘ down " (a clockwise quarter-turn) — same spot as + 27 0 ∘ Keep turning past 36 0 ∘
Intuition Why negative angles land in the same placesA clockwise turn of 9 0 ∘ 27 0 ∘ θ = − 9 0 ∘ θ = + 27 0 ∘ same point , hence the same coordinates. Nothing on this page breaks for negative θ
The set of all points at distance 1 (Section 2) from the origin. "Unit" just means "one". Drawn out, it's a perfectly round loop of radius 1
1 5 1 deliberate simplification . The long slant of every triangle we build will equal the radius, so making the radius 1 1 1 1 = 1 = unit circle and the messy constant vanishes.
Definition Cosine and sineSend out the ray at angle θ P by definition :
cos θ = the x -coordinate of P ( "how far right" ) sin θ = the y -coordinate of P ( "how far up" )
That's it — cos sin not magic; they are literally the shadow-on-the-floor and the height of the point on the circle.
Intuition Why these can be negativeIn Quadrant II the point is up-and-left , so its x cos θ < 0 y < 0 sin θ < 0 θ θ = − 9 0 ∘ x = 0 y = − 1 cos ( − 9 0 ∘ ) = 0 sin ( − 9 0 ∘ ) = − 1 cos sin which quadrant the point sits in — the same sign table from Section 1. This is the whole reason the parent note keeps saying "the quadrant decides the sign".
sin 2 θ sin ( θ 2 ) Why it tempts you: the little 2 θ Truth: sin 2 θ ( sin θ ) 2 first take the sine, then square the result. Completely different from sin ( θ 2 )
2 a 2 a × a a 2 area of a square whose side has length a 3 2 = 3 × 3 = 9 1 2 = 1 × 1 = 1
Definition Right triangle & PythagorasA right triangle is a triangle with one 9 0 ∘ legs ; the long slanted side opposite it is the hypotenuse . Pythagoras Theorem says:
( leg 1 ) 2 + ( leg 2 ) 2 = ( hypotenuse ) 2 . 2
looks like
Build a square on each side of the triangle (see figure). The two squares on the legs, glued together, have exactly the same total area as the big square on the hypotenuse. That area-balance is the whole theorem — and it is the same rule we used to measure distance in Section 2.
Intuition Why this specific tool, here?Our point P = ( cos θ , sin θ ) = ∣ cos θ ∣ = ∣ sin θ ∣ = = 1 one law that relates the two legs to the hypotenuse , and relating "how far right" to "how far up" to "distance 1 precisely the question this whole chapter asks. No other tool connects those three lengths.
Feeding our triangle into Pythagoras:
∣ cos θ ∣ 2 + ∣ sin θ ∣ 2 = 1 2 ⟹ cos 2 θ + sin 2 θ = 1. This is the master identity the parent note derives — and you now own every symbol in it.
± k k what positive number, squared, gives k The symbol ± two numbers whose square is k
± cos 2 θ = 25 16 cos θ + 5 4 or − 5 4 quadrant (Section 1's sign table) picks the right sign. This is the single most-forgotten step in the worked examples.
To read the other two identities you need four more machines. Each is built purely from sin cos Reciprocal and Quotient trig identities ).
Intuition Why "divide by a leg" gives the other two identitiesThe parent note derives the child identities by dividing the master identity by a squared leg. Divide every term of cos 2 θ + sin 2 θ = 1 cos 2 θ c o s 2 θ c o s 2 θ + c o s 2 θ s i n 2 θ = c o s 2 θ 1 . cos 2 θ cos 2 θ = 1 cos 2 θ sin 2 θ = tan 2 θ cos 2 θ 1 = sec 2 θ 1 + tan 2 θ = sec 2 θ . sin 2 θ s i n 2 θ c o s 2 θ + s i n 2 θ s i n 2 θ = s i n 2 θ 1 ⟹ cot 2 θ + 1 = csc 2 θ , sin 2 θ cos 2 θ = cot 2 θ sin 2 θ 1 = csc 2 θ 1 + cot 2 θ = csc 2 θ . exactly the pieces that pop out when you divide the master identity.
Common mistake Forgetting these are undefined sometimestan θ sec θ cos θ blow up when cos θ = 0 (i.e. θ = 9 0 ∘ , 27 0 ∘ cot , csc sin θ = 0 1 + tan 2 θ = sec 2 θ
Coordinate plane x, y axes
Distance = slant of right triangle
Angle theta as a turn, plus or minus
cos = x-coord, sin = y-coord
Right triangle and Pythagoras
1 + tan^2 = sec^2 and 1 + cot^2 = csc^2
plus or minus square root
Cover the right side. If any answer surprises you, reread that section before the parent note.
What are the x y The horizontal ("how far right") and vertical ("how far up") number-lines crossing at the origin
How do you measure the straight-line distance from origin to ( x , y ) x × x + y × y — the slant of a right triangle with legs
x and
y What does θ A turn from the positive x
What unit does this page use for angles, and how is it marked? Degrees, always written with the ∘
Where does θ = − 9 0 ∘ Straight down; same point as θ = + 27 0 ∘
What is the unit circle? All points at distance 1
Define cos θ sin θ cos θ x sin θ y
In which quadrants is cos θ Quadrants II and III (point is to the left, x < 0
In which quadrants is sin θ Quadrants III and IV (point is below the axis, y < 0
What does the superscript 2 cos 2 θ ( cos θ ) 2
Why does squaring make the identity work in every quadrant? Squaring erases minus signs, so cos 2 θ cos θ
State Pythagoras' theorem leg² + leg² = hypotenuse²
Why is the hypotenuse equal to 1 It is the radius of the unit circle, which is 1
Write tan θ sec θ csc θ cot θ sin cos tan = sin / cos sec = 1/ cos csc = 1/ sin cot = cos / sin
Get 1 + tan 2 θ = sec 2 θ Divide every term of cos 2 + sin 2 = 1 cos 2 θ
When are tan θ sec θ When cos θ = 0 θ = 9 0 ∘
Why does solving cos 2 θ = k ± Two numbers square to k
