G1: Trigonometric functions, identities and equations

Sine and cosine extend to angles of any magnitude through the unit circle. Tangent is $\frac{\sin\theta}{\cos\theta}$, while cosecant, secant, and cotangent are the corresponding reciprocals. Inverse-trigonometric notation returns a principal value, not every possible angle.

Memorise or derive the exact sine, cosine, and tangent values for $30^\circ$, $45^\circ$, and $60^\circ$, equivalently $\frac\pi6$, $\frac\pi4$, and $\frac\pi3$. Use signs and symmetry to extend them to other quadrants.

For sine and cosine, the magnitude of the vertical multiplier gives the amplitude. A horizontal multiplier compresses the period, while a horizontal divisor stretches it. A vertical constant shifts the centre line. Tangent has no amplitude and has vertical asymptotes where it is undefined.

Use symmetry and periodicity to sketch the required interval, marking intercepts, extrema, centre lines, and asymptotes. The syllabus includes forms with either a horizontal multiplier or divisor, so read the argument before determining the period.

Reciprocal, quotient, Pythagorean, compound-angle, and double-angle identities let one expression be rewritten into another. In a proof, start from one side and transform it into the other. To combine $a\cos\theta+b\sin\theta$, choose an R-form whose expanded coefficients match $a$ and $b$, with $R=\sqrt{a^2+b^2}$.

Rearrange to a known trigonometric value, find a reference or principal angle, then use the relevant quadrants and period to list all solutions in the stated interval. Verify endpoints and exclude values where the original expression is undefined. A general solution is outside this syllabus requirement.

Periodic motion, tides, and rotating systems may be modelled with sine or cosine. Interpret amplitude, centre line, period, and phase from the context, and use the stated angle unit consistently.