Theorem

How the ratios move across the first quadrant

Trigonometry • Trigonometric Identity

Statement

θ

increases from

0^{\circ}

9 0^{\circ}

sin θ

increases strictly from

0

1

tan θ

increases strictly from

0

without bound, and

cos θ

decreases strictly from

1

0

; consequently

sec θ

increases while

cosec θ

and

cot θ

decrease.

How to use it

1Compare two acute angles with no table: the larger angle has the larger sine and tangent and the smaller cosine.
2Use it for uniqueness - a strictly monotonic ratio takes each value at most once on $(0^{\circ}, 9 0^{\circ})$ , so an equation such as $tan x = 1$ has exactly one acute solution.
3Bracket an awkward angle between two standard angles and settle the comparison from the direction of movement alone.

Which is larger,

cos 2 8^{\circ}

cos 3 5^{\circ}

Step-by-step

1
$2 8^{\circ} < 3 5^{\circ}$ , both acute
Why: Both lie in the first quadrant, where the rule applies.
2
$cos decreases on (0^{\circ}, 9 0^{\circ}) \Rightarrow cos 2 8^{\circ} > cos 3 5^{\circ}$

Answer

cos 2 8^{\circ}

Watch out

The directions hold on the first quadrant only; past

9 0^{\circ}

the sine begins to fall and the cosine changes sign.

tan θ

is unbounded, so 'increasing' here does not mean it tends to a finite limit - it is undefined at

9 0^{\circ}

Also known as

sine increases cosine decreases, monotonicity of the trigonometric ratios, comparing two trigonometric values, increasing and decreasing trig ratios