Intuition The one core ideaA right triangle has one square corner, so its other two angles must split the leftover 90° complements . Because the side that is "in front of" one angle is "beside" the other, swapping which angle you stand at swaps opposite and adjacent , and that single swap is the entire reason behind every co-function identity .
This page assumes you have seen nothing . We build every letter, every symbol, and every picture the parent note leans on, one careful brick at a time.
Before we can even say "angle" we need three plainer words.
Definition Point, line segment, corner
A point is a single dot — a location with no size, like the tip of a sharp pencil.
A line segment is the straight path joining two points — think of a pulled-taut string between two pins.
A corner (or vertex ) is where two line segments meet at a point.
Now the angle is easy to say:
An angle is the amount of turning between two line segments that meet at a corner. We measure "how open" the corner is.
Definition The angle mark ∠ and how we name anglesThe little curved mark ∠ is drawn inside a corner to point at the angle. To name an angle we list three points: ∠AOB means "the angle at the middle-listed point O A B
We name angle sizes with Greek letters because plain letters like a , b , c
Worked example What to see in the figure aboveTwo arms leave the vertex O θ between the arms — that is exactly what ∠AOB points at.
Definition Degree and the symbol° 360 equal slices. Each slice is one degree , written with a tiny circle: 1° 90°
Why 360? History and convenience (360 divides evenly by lots of numbers). What matters here: the square corner is 90° , and that number is the hero of this whole chapter.
90° 90°
There is a second way to measure angles using 2 π 90° Radian Measure 90° 2 π same square corner. We give a concrete conversion in Section 6.
Definition Triangle and right triangleA triangle is a shape made of three points (corners) joined by three line segments (sides). A right triangle is one whose corners include a 90°
Definition The right-angle mark □Whenever a corner is exactly 90° not draw the curved ∠ arc. Instead we draw a small square box □ tucked into the corner. Seeing □ in a figure is a promise: "this corner is precisely 90°
Now the three sides. Their names are not fixed to the drawing — they depend on which angle you are looking from . Stand at angle θ
Definition Opposite, Adjacent, Hypotenuse
Hypotenuse (h Opposite (opp) — the side facing your angle θ Adjacent (adj) — the side beside your angle θ
Worked example What to see in the figure aboveThe plum □ at corner C 90° θ A beside you (adjacent), the orange side is across from you (opposite), and the plum-labelled slanted side is the hypotenuse.
Intuition The picture to burn in"Opposite" = the wall across the room from you. "Adjacent" = the wall right next to you. Move to the other corner and the two walls trade names . Hold that image — it is the seed of everything.
This vocabulary is developed fully in Right Triangle Trigonometry (SOH-CAH-TOA) .
We now meet the functions. A function is a machine: feed in an angle, get out a number.
Definition The six trig functions (from SOH-CAH-TOA)For angle θ sin θ = hyp opp , cos θ = hyp adj , tan θ = adj opp csc θ = s i n θ 1 = opp hyp , sec θ = c o s θ 1 = adj hyp , cot θ = t a n θ 1 = opp adj
The capitals are just the first letters of the lowercase words:
S in = O pp/ H yp, C os = A dj/ H yp, T an = O pp/ A dj.
"SOH" is exactly sin = hyp opp
θ acute here
These ratios come from a real right triangle, and a triangle can only exist if each of its two non-right corners is more than 0° 90° — such an angle is called acute . So throughout the triangle picture, θ ϕ acute : 0° < θ < 90°
Why these ratios and not the raw lengths? Because a ratio forgets the size of the triangle. Two triangles with the same angles but different sizes give the same sin θ shape (the angle) alone. That is exactly what we want when studying angles.
Definition Reading the notationsin θ sin θ apply the sine machine to the angle θ ." The θ one inseparable thing — you can not split it like multiplication. (For example, sin ( 90° − θ ) not sin 90° − sin θ
Definition The superscript-2 notationsin 2 θ sin 2 θ ( sin θ ) 2 sin θ 2
Here is the geometric fact everything hinges on.
Intuition WHY do a triangle's angles add to180° one half-turn — that is 180° flat straight line , and a straight line is 180° any triangle always sum to 180°
Subtract the 90° θ + ϕ = 90° ⟹ ϕ = 90° − θ . θ 0° < θ < 90° ϕ = 90° − θ
Definition Complementary anglesTwo angles are complementary when they add up to 90° θ 90° − θ complement . (Contrast with supplementary , which means adding to 180° Complementary and Supplementary Angles .)
Worked example What to see in the figure aboveThe plum □ at C 90° θ A ϕ B remaining 90°
Intuition Why the lettersθ 90° − θ always θ 90° − θ
Now combine Sections 2 and 4. Stand at θ ϕ = 90° − θ
The side that was opposite θ adjacent to ϕ
The side that was adjacent to θ opposite ϕ
The hypotenuse never moves.
Worked example What to see in the figure aboveThe same highlighted side is orange on the left panel (labelled "opp of θ ϕ is the whole proof.
So the same physical side gets read two ways:
sin θ = h opp of θ , cos ( 90° − θ ) = cos ϕ = h adj of ϕ . same top and bottom — they are the same number. That is the whole engine:
sin θ = cos ( 90° − θ )
You need no more than this picture. The parent note runs the algebra; you now see why it must be true.
Definition The blackboard end-marks
■ ⇒ therefore " or "which forces."⟺
2 π π pie ") is a fixed number (≈ 3.14159 radian measure a half-turn is π 2 π 90° 2 π Radian Measure .
Worked example Concrete degree ↔ radian conversionThe bridge is 180° = π 180° π degrees → radians :
90° = 90 × 180 π = 2 π , 60° = 60 × 180 π = 3 π . π 180° 4 π = 4 π × π 180° = 45°
Point line corner then angle
Degree and the 90 square corner
Right triangle with square corner
Opposite Adjacent Hypotenuse naming
Six trig ratios SOH CAH TOA
Two acute angles complementary
phi equals 90 minus theta
Opp and Adj swap between the two angles
Cover the right side and answer aloud. If any stalls, re-read its section above.
What does the symbol θ θ O
What is one degree, and how many make a square corner? A degree is 360 1 90°
What does the small box □ in a corner tell you? That corner is exactly 90°
Which side of a right triangle is the hypotenuse? The longest side, directly across from the 90°
Why must θ A triangle corner must be between 0° 90°
What happens to sin θ cos θ tan θ θ → 90° sin → 1 cos → 0 tan
When you move from angle θ They swap — the opposite of one becomes the adjacent of the other.
Write the three primary ratios of SOH-CAH-TOA. sin = hyp opp cos = hyp adj tan = adj opp
Does sin 2 θ sin ( θ 2 ) No — it means ( sin θ ) 2 sin θ
Two angles are complementary when they add to what? 90° = 180°
Why do a triangle's three angles sum to 180° Walking round it turns you a half-turn; equivalently the three torn corners fit into a straight line (180°
Convert 60° 60 × 180 π = 3 π
Co-function identities — the parent this page prepares you for.Right Triangle Trigonometry (SOH-CAH-TOA) — full home of opp/adj/hyp and the six ratios.Complementary and Supplementary Angles — the 90° 180° Radian Measure — the π /2 Pythagorean Identities — the next tool (sin 2 + cos 2 = 1 Hinglish version → 
