Coaching Centre
Exact values and obtuse angles
Exact trigonometric values for 30°, 45° and 60° from special triangles, and sine, cosine and tangent of obtuse angles.
Exact values
| $30^\circ$ | $45^\circ$ | $60^\circ$ | |
|---|---|---|---|
| $\sin$ | $\tfrac{1}{2}$ | $\tfrac{1}{\sqrt{2}}$ | $\tfrac{\sqrt{3}}{2}$ |
| $\cos$ | $\tfrac{\sqrt{3}}{2}$ | $\tfrac{1}{\sqrt{2}}$ | $\tfrac{1}{2}$ |
| $\tan$ | $\tfrac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ |
Obtuse angles
Find exact values of $\sin 150^\circ$ and $\cos 120^\circ$.
$\sin 150^\circ = \sin 30^\circ = \tfrac{1}{2}$
$\cos 120^\circ = -\cos 60^\circ = -\tfrac{1}{2}$
$sin 60^circ$
$\tfrac{\sqrt{3}}{2}$
$ an 30^circ$
$\tfrac{1}{\sqrt{3}}$
$cos(180^circ - heta)$
$-\cos\theta$
Exact value of $\sin 150^\circ$?
$\sin 30^\circ$.
Exact value of $\cos 120^\circ$?
$-\cos 60^\circ$.
$\tan 45^\circ$?
Half a square.
$\sin 60^\circ$ equals:
From the $1$, $\sqrt{3}$, $2$ triangle.
$\tan 135^\circ$?
$-\tan 45^\circ$.
Solve $\sin\theta = 0.5$ for $0^\circ \le \theta \le 180^\circ$.
$30^\circ$ and $180 - 30$.
Solve $\cos\theta = -\tfrac{1}{\sqrt{2}}$ for $0^\circ \le \theta \le 180^\circ$.
$180 - 45$.
For an obtuse angle, which ratio is positive?
$\sin(180 - \theta) = \sin\theta$.
New numbers every time. Keep going until you get five in a row.