E6.3Extended tier0580 · Section 6: Trigonometry

Exact trigonometric values

Key learnings

Know exact trig values for special angles
For $0°, 30°, 45°, 60°, 90°$ the sine and cosine take exact (often surd) values, and tan is exact for $0°, 30°, 45°, 60°$ . These are needed when a question says "give an exact value" — a rounded decimal will not earn the mark.
The values come from two special triangles
The $45°$ - $45°$ - $90°$ triangle (sides $1,1,\sqrt2$ ) gives $\sin45°=\cos45°=\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}$ and $\tan45°=1$ . The $30°$ - $60°$ - $90°$ triangle (sides $1,\sqrt3,2$ ) gives $\sin30°=\cos60°=\tfrac12$ , $\cos30°=\sin60°=\dfrac{\sqrt3}{2}$ , $\tan30°=\dfrac{1}{\sqrt3}$ and $\tan60°=\sqrt3$ .
sin and cos swap for complementary angles
Because $30°+60°=90°$ , $\sin30°=\cos60°$ and $\cos30°=\sin60°$ . Likewise $\sin0°=\cos90°=0$ and $\cos0°=\sin90°=1$ . This symmetry halves how much you must memorise.

Common mistakes

Drawn from real examiner reports.

Confusing sin and cos of 30° and 60°

A very common slip is writing $\sin30°=\dfrac{\sqrt3}{2}$ (which is actually $\sin60°$ ). Remember $\sin30°=\tfrac12$ is the small value because $30°$ is a small angle, and $\sin60°=\dfrac{\sqrt3}{2}\approx0.87$ is the larger one.

Giving a decimal instead of an exact value

Writing $\tan60°=1.73$ instead of $\sqrt3$ , or $\sin45°=0.707$ instead of $\dfrac{\sqrt2}{2}$ , loses the mark when the question asks for an exact value. Leave surds as surds.

Squaring √3/2 as √3/4 instead of 3/4

When squaring a surd value, square top and bottom: $\left(\dfrac{\sqrt3}{2}\right)^2=\dfrac{3}{4}$ , not $\dfrac{\sqrt3}{4}$ . Here $(\sqrt3)^2=3$ , so the numerator becomes $3$ .

Leaving 1/√2 un-rationalised

If a question asks for the form $\dfrac{\sqrt2}{2}$ or $\dfrac{\sqrt3}{3}$ , don't stop at $\dfrac{1}{\sqrt2}$ or $\dfrac{1}{\sqrt3}$ — rationalise the denominator to reach the requested surd form.

Assuming tan 90° has a value

$\tan90°$ is undefined (since $\cos90°=0$ , tan would divide by zero), so there is no exact value to quote. The syllabus requires exact tan only for $0°, 30°, 45°, 60°$ .

Exam tips

Substitute exact values, then simplify
In formulas like the area $\tfrac12 ab\sin C$ , put in the exact value — e.g. $\sin30°=\tfrac12$ — before evaluating. This keeps answers exact and avoids rounding, e.g. $\tfrac12\times6\times8\times\tfrac12=12$ .
Rationalise surd denominators
When a value has a surd in the denominator, rationalise it: $\tan30°=\dfrac{1}{\sqrt3}=\dfrac{\sqrt3}{3}$ and $\sin45°=\dfrac{1}{\sqrt2}=\dfrac{\sqrt2}{2}$ . This is usually the form the mark scheme wants.
Rebuild the values from two triangles
If you forget a value, sketch the two triangles: legs $1,1,\sqrt2$ for $45°$ , and half an equilateral ( $1,\sqrt3,2$ ) for $30°$ and $60°$ . Read sin, cos and tan straight off the sides.

Revision notes

For the special angles, sine and cosine take exact values; tan is exact for $0°$ to $60°$ ( $\tan90°$ is undefined).

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

Flashcards

Foundation

What is the exact value of tan 45°?

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Exact trigonometric values

Key learnings

Common mistakes

Exam tips

Revision notes

The values you must know

Where they come from — two special triangles

Helpful symmetry

Don't get caught out

Flashcards

Practice questions