Quadrant, sign and related acute angle

Use the ASTC rule and the related acute angle to find the exact value of sin, cos and tan of any angle, including negative angles and angles over 360°.

Which functions are positive?

Quadrant 2 ($90^\circ$ to $180^\circ$): Sin positive

Quadrant 3 ($180^\circ$ to $270^\circ$): Tan positive

Quadrant 1 ($0^\circ$ to $90^\circ$): All positive

Quadrant 4 ($270^\circ$ to $360^\circ$): Cos positive

Remember it anticlockwise from quadrant 1 as All Stations To Central.

The is the acute angle between the ray and the $x$-axis.

Quadrant	Angle $\theta$	Related angle
2	$180^\circ - \alpha$	$\alpha$
3	$180^\circ + \alpha$	$\alpha$
4	$360^\circ - \alpha$	$\alpha$

Related angles

Angles past 360° or below 0° land on the same rays as angles from 0° to 360°.

Try this

Set $\theta = 150^\circ$, $210^\circ$ and $330^\circ$. What do they have in common?
Set $\theta = -30^\circ$. Which angle between $0^\circ$ and $360^\circ$ is the same ray?

Example 1●●

Find the exact value of $\cos 210^\circ$.

Quadrant 3, where only tan is positive, so cos is negative.

210° is between 180° and 270°.

Related angle $30^\circ$.

210 = 180 + 30.

$\cos 210^\circ = -\cos 30^\circ = -\dfrac{\sqrt{3}}{2}$

Example 2●●

Find the exact value of $\tan(-120^\circ)$.

$-120^\circ$ is in quadrant 3, where tan is positive.

Going clockwise 120° lands on the same ray as 240°.

Related angle $60^\circ$, so $\tan(-120^\circ) = \tan 60^\circ = \sqrt{3}$.

Flashcards

ASTC

Q1 all, Q2 sin, Q3 tan, Q4 cos are positive.

Related angle of 180° + α

$\alpha$, in quadrant 3.

sin(180° − θ)

$\sin\theta$

cos(−θ)

$\cos\theta$

Practice

$\sin 150^\circ =$
$\cos 300^\circ =$
$\tan 135^\circ =$
What is the related acute angle of $320^\circ$?
$\sin\theta < 0$ and $\cos\theta > 0$. Which quadrant is $\theta$ in?
Find the exact value of $\sin 225^\circ$. (Type something like -sqrt(3)/2.)
Find the exact value of $\cos 420^\circ$.
Which equals $\cos 20^\circ$?

Which functions are positive?

The related acute angle