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G5.20 Exact Trigonometric Values

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01 — Explanation

Why do I need exact values if calculators are allowed in the exam?

An "exact value" is a precise expression using fractions and surds (e.g. $\dfrac{\sqrt3}{2}$ ), not a rounded decimal (e.g. $0.866$ ). Questions phrased "work out the exact value" or "show that..." are specifically designed so a calculator's decimal output cannot earn full marks — you must recall and manipulate the exact forms directly.

Exam Tip

Writing a decimal answer when "exact value" is requested scores zero, even if the decimal is numerically correct — the question is testing recall of the surd/fraction form itself.

02 — Explanation

Where do the values for 30°, 45° and 60° actually come from?

They're derived from two special triangles, not memorised arbitrarily:

45-45-90 (half a unit square, cut along the diagonal): sides $1,1,\sqrt2$ . This gives $\sin45°=\cos45°=\dfrac{\sqrt2}{2}$ and $\tan45°=1$ .
30-60-90 (half an equilateral triangle, side 2, bisected): sides $1,\sqrt3,2$ . This gives:

Angle	$\sin$	$\cos$	$\tan$
$30°$	$\frac12$	$\frac{\sqrt3}{2}$	$\frac{1}{\sqrt3}=\frac{\sqrt3}{3}$
$45°$	$\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$	$1$
$60°$	$\frac{\sqrt3}{2}$	$\frac12$	$\sqrt3$
$0°$	$0$	$1$	$0$
$90°$	$1$	$0$	undefined

Exam Tip

A fast way to recall the table: write $\sqrt0,\sqrt1,\sqrt2,\sqrt3,\sqrt4$ over 2 — that's $\sin(0°,30°,45°,60°,90°)$ in order. Cosine is the same sequence reversed.

03 — Explanation

How do I use exact values to find an exact side length?

Set up the ratio exactly as in any SOHCAHTOA problem, but substitute the exact value instead of a decimal.

Example: Hypotenuse $=16\text{ cm}$ , angle $30°$ , find the opposite side exactly.

\sin30°=\frac{x}{16} \;\Rightarrow\; \frac12=\frac{x}{16} \;\Rightarrow\; x=8\text{ cm}

If the result contains a surd in the denominator, rationalise it: $\dfrac{12}{\sqrt3}=\dfrac{12}{\sqrt3}\times\dfrac{\sqrt3}{\sqrt3}=\dfrac{12\sqrt3}{3}=4\sqrt3$ .

04 — Explanation

Why is \tan(90°) undefined rather than 0 or infinite?

Using $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ : at $90°$ , $\sin90°=1$ and $\cos90°=0$ , giving $\dfrac10$ — division by zero, which has no defined value. "Undefined" is the only correct answer; stating $0$ or claiming it's infinite both lose the mark.

05 — Explanation

How do exact values combine with other topics, like area?

Example: An equilateral triangle has sides of $10\text{ cm}$ . Find its exact area.

Bisecting it gives a 30-60-90 triangle with hypotenuse $10$ and base $5$ .
Height: $\sin60°=\dfrac{h}{10} \Rightarrow h=10\times\dfrac{\sqrt3}{2}=5\sqrt3\text{ cm}$ .
Area $=\tfrac12\times10\times5\sqrt3=25\sqrt3\text{ cm}^2$ .

Exam Tip

Surds can only combine directly if the number under the root is identical (e.g. $4\sqrt3+2\sqrt3=6\sqrt3$ ) — $\sqrt2+\sqrt3$ cannot be simplified to $\sqrt5$ , a common and incorrect shortcut.

06 — Explanation

How do I avoid mixing up the two special triangles?

A quick way to keep them separate: the 45-45-90 triangle has two equal angles and its ratio involves $\sqrt{\mathbf2}$ ; the 30-60-90 triangle has three different angles and its ratio involves $\sqrt{\mathbf3}$ . If you're ever unsure mid-exam, it only takes a few seconds to sketch both triangles from scratch — a square cut along its diagonal, and an equilateral triangle cut by its height — and read the ratios straight off, rather than relying purely on memory.

Worked Example

Solve $2\cos(x)=\sqrt3$ for $0°\le x\le90°$ .

Isolate the trig function: $\cos(x)=\dfrac{\sqrt3}{2}$ .
Recall which angle gives this exact cosine value: $\cos30°=\dfrac{\sqrt3}{2}$ .
Answer: $x=30°$ .