The Pythagorean identity is
It is true for every angle for which sine and cosine are defined. In IB Mathematics: Applications and Interpretation, it is HL-only content in .
The Pythagorean identity is
It is true for every angle for which sine and cosine are defined. In IB Mathematics: Applications and Interpretation, it is HL-only content in .
Here, means , not . The same convention applies to .
On the unit circle, a point at angle from the positive horizontal axis has coordinates
The unit circle has radius , so its equation is
Substituting and gives
Therefore, the identity is Pythagoras' theorem applied to the horizontal and vertical coordinates of a point on the unit circle. It is an identity because it is true for all permitted values of , rather than only particular solutions.
| Known quantity | Rearranged identity |
|---|---|
These rearrangements let you replace one squared trigonometric function with the other. This is especially useful when simplifying an expression or rewriting an equation so that it contains only sine or only cosine. However, after taking a square root, you must write both possible signs first. The identity determines the magnitude, while the stated quadrant, interval, or graph determines whether the value is positive or negative.
Suppose and lies in the third quadrant. Then
Hence,
Because sine is negative in the third quadrant, .
The identity can also simplify expressions. For example,
A common misconception is to write . This is false: both terms must be squared. Another common error is taking only the positive square root; the quadrant determines the sign.
In an AI HL response, state the identity, substitute the known value, and justify the sign using the quadrant or unit circle. When asked to solve, use your GDC graphically and report every solution in the stated interval. Keep the calculator in radian mode unless the degree symbol is shown.