State and apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, and use it together with related identities to simplify expressions and solve equations

What this dot point is asking

QCAA wants you to know the Pythagorean identity and to apply it together with the quadrant rules (from the unit circle) to find missing trig values, simplify trig expressions, and solve trig equations.

The Pythagorean identity

For any angle $\theta$:

This comes directly from the unit-circle definition: a point at angle $\theta$ on the unit circle has coordinates $(\cos\theta, \sin\theta)$, and these satisfy $x^2 + y^2 = 1$.

The tangent identity

Dividing the Pythagorean identity by $\cos^2 \theta$ gives:

(QCAA Methods uses this less than the basic form, but it appears occasionally.)

Why the identity matters for solving equations

An equation mixing $\sin\theta$ and $\cos\theta$ (or their squares) usually cannot be solved directly, because two different functions of the same angle appear. The Pythagorean identity lets you replace $\cos^2\theta$ with $1 - \sin^2\theta$ (or vice versa) so that only one function remains, turning the equation into a quadratic in that single function. Factoring or the quadratic formula then gives the values of that function, and the unit circle converts each value into all the angles in the required interval. This reduce-to-one-function strategy is the most common identity application in Methods.

Standard manipulations

Express in one function

$\sin^2 \theta = 1 - \cos^2 \theta$, and $\cos^2 \theta = 1 - \sin^2 \theta$. Useful for rewriting an expression in a single trig function before solving.

Difference of squares

$1 - \sin^2 \theta = (1-\sin\theta)(1+\sin\theta) = \cos^2\theta$.

Combine over a common denominator

$\frac{1}{\cos\theta} - \cos\theta = \frac{1 - \cos^2\theta}{\cos\theta} = \frac{\sin^2\theta}{\cos\theta} = \sin\theta \tan\theta$.

Given one trig value, find the others

Use the Pythagorean identity to compute the magnitude, then use the quadrant to fix the sign.

Example: $\cos\theta = -7/25$ and $\theta$ is in Q3.

$\sin^2 \theta = 1 - 49/625 = 576/625$.

$\sin\theta = \pm 24/25$. In Q3, sine is negative, so $\sin\theta = -24/25$.

$\tan\theta = (-24/25) / (-7/25) = 24/7$ (positive, as expected in Q3).

Solving trig equations using identities

If an equation contains both $\sin$ and $\cos$ (or $\sin^2$ and $\cos^2$), use the identity to reduce it to one function.

Example: solve $2 \cos^2 \theta - \sin\theta - 1 = 0$ for $\theta \in [0, 2\pi]$.

Replace $\cos^2 \theta = 1 - \sin^2 \theta$:

$2(1 - \sin^2 \theta) - \sin\theta - 1 = 0$

$1 - 2 \sin^2 \theta - \sin\theta = 0$

$2 \sin^2 \theta + \sin\theta - 1 = 0$

Factor: $(2\sin\theta - 1)(\sin\theta + 1) = 0$.

$\sin\theta = 1/2 \implies \theta = \pi/6$ or $5\pi/6$.

$\sin\theta = -1 \implies \theta = 3\pi/2$.

Three solutions in the stated interval.