Questions that say "without using a calculator" or ask for exact values expect the special-angle ratios. All of them come from two right-angled triangles.
Half of an equilateral triangle of side \(2\) gives the \(30^\circ\) and \(60^\circ\) ratios.
A right-angled isosceles triangle with legs \(1\) gives the \(45^\circ\) ratios.
Exact values for \(30^\circ\), \(45^\circ\) and \(60^\circ\)
| Ratio | \(30^\circ\) / \(\dfrac{\pi}{6}\) | \(45^\circ\) / \(\dfrac{\pi}{4}\) | \(60^\circ\) / \(\dfrac{\pi}{3}\) |
|---|---|---|---|
| \(\sin\theta\) | \(\dfrac{1}{2}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(\cos\theta\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{2}\) |
| \(\tan\theta\) | \(\dfrac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) |
Values on the axes
| Ratio | \(0^\circ\) / \(0\) | \(90^\circ\) / \(\dfrac{\pi}{2}\) | \(180^\circ\) / \(\pi\) | \(270^\circ\) / \(\dfrac{3\pi}{2}\) | \(360^\circ\) / \(2\pi\) |
|---|---|---|---|---|---|
| \(\sin\theta\) | \(0\) | \(1\) | \(0\) | \(-1\) | \(0\) |
| \(\cos\theta\) | \(1\) | \(0\) | \(-1\) | \(0\) | \(1\) |
| \(\tan\theta\) | \(0\) | undefined | \(0\) | undefined | \(0\) |
A squared value has two possible signs — choose using the angle's position. For an acute angle every ratio is positive, so \((\tan\theta)^2=\dfrac{1}{3}\) with \(\theta\) acute gives \(\tan\theta=+\dfrac{1}{\sqrt{3}}\).
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