Trigonometry and identities in N(A)-Level Additional Mathematics (4051): the sine, cosine and tangent ratios on the unit circle and their signs by quadrant, the quotient and Pythagorean identities, and solving trigonometric equations within a stated range

Trigonometry beyond the triangle

Trigonometry in the Geometry and Trigonometry strand of N(A)-Level Additional Mathematics (SEAB 4051) extends the sine, cosine and tangent ratios from right-angled triangles to any angle, using the unit circle. From that foundation come the two identities and the method for solving trigonometric equations across a range. The unit circle is the picture that holds the whole topic together.

This overview links three dot points: the trigonometric ratios and the unit circle, the identities, and solving trigonometric equations. Each has its own worked answers and practice.

The ratios and the unit circle

The trigonometric ratios and the unit circle outcome defines the ratios for any angle. A point on the unit circle at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$, so cosine is the $x$-coordinate, sine is the $y$-coordinate, and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. The sign of each ratio depends on the quadrant, following the ASTC pattern: All positive in the first quadrant, then Sine, Tangent and Cosine in turn.

The trigonometric identities

The trigonometric identities outcome uses two relationships. The quotient identity is $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, and the Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$. Rearranging the second gives $\sin^2\theta = 1 - \cos^2\theta$, which is the standard substitution when simplifying an expression or proving a result.

Solving trigonometric equations

The solving trigonometric equations outcome finds all angles in a stated range. Find the basic (reference) angle from the positive value of the ratio, then use the quadrant signs to place a solution in each quadrant where the ratio has the required sign, listing every angle within the range.

How this module is examined

• Both papers, all questions. Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, 50 percent) cover the full syllabus, and you answer every question.
• List every solution in the range. A trigonometric equation usually has more than one solution in the stated range; use the quadrant rule to find them all, not just the first.
• Round angles to 1 decimal place. Unless told otherwise, give a non-exact angle in degrees to 1 decimal place, as the syllabus requires.

Check your knowledge

Short questions across the three outcomes. Work them with full method, then check the solutions.

1. State the exact value of $\sin 30^\circ$. (1 mark)
2. In which quadrant are both $\sin\theta$ and $\cos\theta$ negative? (1 mark)
3. Simplify $\sin^2\theta + \cos^2\theta$. (1 mark)
4. Solve $\cos\theta = 0.5$ for $0^\circ \leqslant \theta \leqslant 360^\circ$. (3 marks)