Basic identities
Every identity below is built on this one unit circle. The sign is the identity equals: the two sides are equal for every angle, not just one. The reciprocal functions here are explored as graphs on the Trigonometry Graphs page. Drag the point round the circle to explore.
1 (opposite ÷ adjacent)
In the base triangle the side opposite is and the side next to it is . Since , dividing the two gives the identity.
2
The base triangle is right-angled with legs and and hypotenuse the radius . Pythagoras gives the identity at once.
3
Standing on the same base radius (length ), the tangent triangle is the base triangle scaled up by . Its hypotenuse, the secant, is therefore .
4
Run the radius up to the top tangent line. This makes the same triangle turned a quarter-turn: the angle now sits at the far end and at the centre. It is the base triangle scaled by , so the hypotenuse (the cosecant) is .
5
The cotangent is the short side of that same turned-over triangle. Compared with the tangent triangle the ratio is simply inverted, which is why it equals both and .
6
Pythagoras on the tangent triangle (legs and , hypotenuse ). It is also divided through by .
7
Pythagoras on the turned-over cotangent triangle (legs and , hypotenuse ). It is also divided through by .
The three similar triangles
Start with the Base Unit Triangle. Divide every side by cosθ to get the tangent triangle; divide every side by sinθ to get the cotangent triangle (turned over, so θ is at the far end). The reciprocal definitions and both Pythagorean identities all read straight off these.
Compound angle formulas
Slide A and B. Each formula is proved from the one lettered diagram, then the rest follow by substitution and division.
Both minus formulas come straight from the plus ones: replace with and use , .
Tangent is sine over cosine:
Now divide every term, top and bottom, by :
so , and with , .
Double angle formulas
Starting with the compound angle formulas and replacing , we can derive the double angle formulas.
Set in :
The two terms are identical, so
Set in :
which gives the basic form
Two more forms follow by using on one term, with the proof beside each:
Set in :
collect the terms:
Harmonic form: a sin x + b cos x
Any is a single shifted wave. Choose a form whose signs match, expand it underneath the original, compare the coefficients, then read and off a right-angled triangle. Faded forms would force a negative length, so they are turned off for these values.
Sketching, maximum and minimum
Which formula do I choose?
There are four forms: and . Pick the one whose expansion has the same + / − pattern as your expression, so both compared coefficients come out positive. That keeps positive and acute (a real triangle, never a negative side).
- → or
- →
- →
- both negative → take out first, then choose as above
The buttons above fade out any form that would need a negative length for the current a and b, so a valid form is always picked for you.
Small angle approximations
When is small and measured in radians, , and . Tap any combination below to compare them on one graph, then drag the zoom: the dots start together at and pull apart as grows.
Why radians and not degrees?
The approximation says the curve and the line almost coincide near . Below is the same picture with on the axis first in radians and then in degrees, each with the line drawn on top.
In radians the line sits right on near 0. In degrees the same line shoots off the top almost at once and is nowhere near the curve, so the approximation only works in radians.