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Cambridge IGCSE Mathematics — 0580 Extended

Topic 6.4: Trigonometry — Trigonometric Functions

\(y = \sin x\) for \(0^\circ \le x \le 360^\circ\)

One full wave. It starts at \(0\), peaks at \(1\) when \(x = 90^\circ\), returns to \(0\) at \(180^\circ\), reaches \(-1\) at \(270^\circ\), and finishes at \(0\) when \(x = 360^\circ\).

Graph of y equals sin x from 0 to 360 degrees with key points at 0, 90, 180, 270 and 360
Sine is positive in the first and second quadrants (\(0^\circ\) to \(180^\circ\)).

\(y = \cos x\)

The same wave as sine, shifted \(90^\circ\) left, so it starts at a peak. \(\cos 0^\circ = 1\), \(\cos 90^\circ = 0\), \(\cos 180^\circ = -1\), \(\cos 270^\circ = 0\), \(\cos 360^\circ = 1\).

Graph of y equals cos x from 0 to 360 degrees starting at 1, crossing zero at 90 and 270, and reaching minus 1 at 180
Cosine is positive in the first and fourth quadrants.

\(y = \tan x\)

Tangent repeats every \(180^\circ\). It has vertical asymptotes at \(90^\circ\) and \(270^\circ\) — those values are undefined. \(\tan 0^\circ = \tan 180^\circ = \tan 360^\circ = 0\), and \(\tan 45^\circ = \tan 225^\circ = 1\).

Graph of y equals tan x from 0 to 360 degrees with dashed vertical asymptotes at 90 and 270 degrees labelled undefined
Never write \(\tan 90^\circ = 1\). The graph shoots to infinity; there is no value.

CAST and related angles

The calculator inverse gives one acute angle \(\alpha\). A full turn almost always has a second solution. CAST (read counterclockwise from the fourth quadrant: Cos, All, Sin, Tan) tells you which quadrants are allowed.

  • Sine positive: \(\alpha\) and \(180^\circ - \alpha\)
  • Cosine positive: \(\alpha\) and \(360^\circ - \alpha\)
  • Tangent positive: \(\alpha\) and \(180^\circ + \alpha\)
CAST circle with All positive in the first quadrant, sine in the second, tangent in the third and cosine in the fourth, plus related-angle rules
From the first quadrant, the same letters read ASTC: All, Sin, Tan, Cos.

Solving equations in \(0^\circ \le x \le 360^\circ\)

Method

  1. Rearrange so that \(\sin x\), \(\cos x\) or \(\tan x\) is the subject.
  2. Find the acute related angle \(\alpha\) from the calculator (or from exact values).
  3. Use CAST to list every solution in the interval. Check the endpoints \(0^\circ\) and \(360^\circ\) if the value is \(0\).

Solve \(\sin x = \dfrac{\sqrt{3}}{2}\) for \(0^\circ \le x \le 360^\circ\).

Unit circle with the line y equals root 3 over 2 intersecting at 60 degrees and 120 degrees
Sine is positive in the first and second quadrants, so \(x = 60^\circ\) or \(x = 180^\circ - 60^\circ = 120^\circ\).

Solve \(2\cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).

Cosine graph from 0 to 360 with the line y equals minus one half intersecting at 120 degrees and 240 degrees
\(\cos x = -\dfrac{1}{2}\). Related angle \(60^\circ\). Cosine is negative in the second and third quadrants: \(180^\circ - 60^\circ = 120^\circ\) and \(180^\circ + 60^\circ = 240^\circ\).

Paper 2 (non-calculator)

When the right-hand side is an exact value from Topic 6.3, \(\alpha\) is one of \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\). You should not need a calculator for \(\sin x = \frac{\sqrt{3}}{2}\) or \(2\cos x + 1 = 0\).

Try this

Solve \(\tan x = 1\) for \(0^\circ \le x \le 360^\circ\).

Show answer
Answer

  1. Related angle: \(\tan 45^\circ = 1\)

    \[ \alpha = 45^\circ \]

  2. Tan is positive in the first and third quadrants

    \[ x = 45^\circ \quad\text{or}\quad x = 180^\circ + 45^\circ = 225^\circ \]

Exam Traps

  • The calculator reports only one solution. In \(0^\circ\) to \(360^\circ\) a sine or cosine equation almost always has two; missing the second loses a mark.
  • Degree mode still applies. \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) should be \(60\), not a small number of radians.

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