When we restricted our attention to rightangled triangles, we only considered the trigonometric ratios $\sin\theta$sinθ, $\cos\theta$cosθ $\tan\theta$tanθ for angles whose size was between $0^\circ$0° and $90^\circ$90°.
The unit circle is a visual tool that allows us to connect the trigonometric ratios to angles of any magnitude. It is defined as a circle with a radius of $1$1 unit centred at the origin on the Cartesian plane. A radius to any point $P$P on the circle is free to move and makes an angle $\theta$θ with the positive horizontal axis. By convention, the angle is measured anticlockwise from the positive horizontal axis. The angle can have any size, positive or negative, depending on how far the point has moved around the circle.
The sine of the angle is defined to be the $y$ycoordinate of point $P$P. For an acute angle $\theta$θ, this makes sense as we have always thought of:
$\sin\theta$sinθ  $=$=  $\frac{\text{opposite}}{\text{hypotenuse}}$oppositehypotenuse 
In the triangle above, this gives us:
$\sin\theta$sinθ  $=$=  $\frac{y\text{coordinate of}\ P}{1}$ycoordinate of P1 
$y\text{coordinate of}\ P$ycoordinate of P  $=$=  $\sin\theta$sinθ 
The cosine of the angle is defined to be the $x$xcoordinate of point $P$P. For an acute angle $\theta$θ, this makes sense as we have always thought of:
$\cos\theta$cosθ  $=$=  $\frac{\text{adjacent}}{\text{hypotenuse}}$adjacenthypotenuse 
In the triangle above, this gives us:
$\cos\theta$cosθ  $=$=  $\frac{x\text{coordinate of}\ P}{1}$xcoordinate of P1 
$x\text{coordinate of}\ P$xcoordinate of P  $=$=  $\cos\theta$cosθ 
The tangent of the angle can be geometrically defined to be $y$ycoordinate of point $Q$Q, where $Q$Q is the intersection of the extension of the line $OP$OP and the tangent of the circle at $\left(1,0\right)$(1,0). Using similar triangles we can define this algebraically as the ratio:
$\frac{\sin\theta}{\cos\theta}$sinθcosθ  $=$=  $\frac{\tan\theta}{1}$tanθ1 
This also represents the gradient of the line that forms the angle $\theta$θ to the positive $x$xaxis and hence this gives us our first trigonometric identity:
$\tan\theta$tanθ  $=$=  $\frac{\sin\theta}{\cos\theta}$sinθcosθ provided $\cos\theta\ne0$cosθ≠0 
Using the applet below, change the angle and take note of the sign (positive or negative) that each ratio has in the four different quadrants.

Depending on where the point is on the unit circle, we say the angle is in one of four quadrants. From the applet, as we watch the point moving through the quadrants, it can be seen that the sine function is positive for angles in the first and second quadrants; cosine is positive for angles in the first and fourth quadrants; and consequently, the tangent is positive for angles in the first and third quadrants.
These can be remembered by having a mental picture of the unit circle diagram and the pattern ASTC: AllSineTangentCosine which refers to which ratios are positive in each quadrant. You might have been taught the mnemonic "All Stations To Central" to help you remember this.
From the applet, we can also see that the values of $\sin\theta$sinθ and $\cos\theta$cosθ are bound between $1$1 and $1$−1. However, the value of $\tan\theta$tanθ is not bound and is undefined at $90^\circ$90° and $270^\circ$270°. The pattern will restart at $360^\circ$360°. We will revisit this when we consider the graphs of the trigonometric functions later in this topic.
We can use symmetry within the circle to find equivalent angles. That is angles with the same trigonometric ratios.
From the diagram we can see that the $y$ycoordinate of point $A$A is the same as point $B$B, hence:
$\sin\theta$sinθ  $=$=  $\sin(180^\circ\theta)$sin(180°−θ) 
We can also see the $x$xcoordinate of $A$A is the same as point $D$D, hence:
$\cos\theta$cosθ  $=$=  $\sin(360^\circ\theta)$sin(360°−θ) 
The $y$ycoordinate of $C$C has the same magnitude but would be negative in comparison to the $x$xcoordinate of $A$A, hence:
$\sin(180+\theta)$sin(180+θ)  $=$=  $\sin\theta$−sinθ 
The trigonometric functions of all angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant. This first quadrant angle may be called the relative acute angle, or the reference angle. The relative acute angle is between $0^\circ$0° and $90^\circ$90°.
We can summarise trigonometric ratios for angles of any magnitude as follows:
First quadrant Angle $\theta$θ $\sin\theta$sinθ is positive $\cos\theta$cosθ is positive $\tan\theta$tanθ is positive 
Second quadrant Angle $180^\circ\theta$180°−θ $\sin(180^\circ\theta)=\sin\theta$sin(180°−θ)=sinθ $\cos(180^\circ\theta)=\cos\theta$cos(180°−θ)=−cosθ $\tan(180^\circ\theta)=\tan\theta$tan(180°−θ)=−tanθ 
Third quadrant Angle $180^\circ+\theta$180°+θ $\sin(180^\circ+\theta)=\sin\theta$sin(180°+θ)=−sinθ $\cos(180^\circ+\theta)=\cos\theta$cos(180°+θ)=−cosθ $\tan(180^\circ+\theta)=\tan\theta$tan(180°+θ)=tanθ 
Fourth quadrant Angle $360^\circ\theta$360°−θ $\sin(360^\circ\theta)=\sin\theta$sin(360°−θ)=−sinθ $\cos(360^\circ\theta)=\cos\theta$cos(360°−θ)=cosθ $\tan(360^\circ\theta)=\tan\theta$tan(360°−θ)=−tanθ 
Express $\cos117^\circ$cos117° in terms of a first quadrant angle.
Think: The angle $117^\circ$117° is between $90^\circ$90° and $180^\circ$180°, so it is in the second quadrant. The point representing $117^\circ$117° on the unit circle diagram, where the radius cuts the circle, must have a negative horizontal coordinate.
Do: Therefore, $\cos117^\circ$cos117° must be the same as $\cos\left(180^\circ117^\circ\right)=\cos63^\circ$−cos(180°−117°)=−cos63°.
Find the exact value of $\cos240^\circ$cos240°.
Think: The angle $240^\circ$240° is in the third quadrant, and remembering ASTC, we know that $\cos240^\circ$cos240° will be negative.
Do: We find its first quadrant reference angle by subtracting $180^\circ$180° to get $60^\circ$60°.
Of course, we remember our exact values for $30^\circ$30°, $45^\circ$45° and $60^\circ$60°! Thus, the exact value of $\cos240^\circ=\cos60^\circ=$cos240°=−cos60°= $\frac{1}{2}$−12.
What other equivalences can you see around the unit circle? What about angles larger than $360^\circ$360° or negative angles?
An angle simply specifies a rotation. A rotation of more than $360^\circ$360° means we have completed more than a full lap around the unit circle. We will land on a point $P$P that we had already passed over. A negative angle means we are rotating in the opposite direction, so we set off clockwise rather than anticlockwise, stopping in a position we are used to thinking about being related to a positive angle between $0^\circ$0° and $360^\circ$360°.
If you think about it, there are literally an infinite number of angles that have the same trigonometric ratio! This behaviour is due to the periodic nature of the trigonometric functions  in basic terms, the values will repeat. So to find the acute reference angle, first, if necessary, add or subtract multiples of $360^\circ$360° to obtain an angle between $0^\circ$0° and $360^\circ$360°. Then decide what quadrant the angle is in as we did before.
Which of the following will have positive answers? Select all correct answers.
$\tan296^\circ$tan296°
$\sin120^\circ$sin120°
$\cos91^\circ$cos91°
$\sin296^\circ$sin296°
$\cos120^\circ$cos120°
$\cos296^\circ$cos296°
$\tan296^\circ$tan296°
$\sin120^\circ$sin120°
$\cos91^\circ$cos91°
$\sin296^\circ$sin296°
$\cos120^\circ$cos120°
$\cos296^\circ$cos296°
For each of the following, rewrite the expression as the trigonometric ratio of a positive acute angle.
You do not need to evaluate the trigonometric ratio.
$\sin93^\circ$sin93°
$\cos195^\circ$cos195°
$\tan299^\circ$tan299°
We want to evaluate $\tan\left(\left(75\right)\right)$tan((−75)) by first rewriting it in terms of the related acute angle $\theta$θ. What is the related acute angle of $\left(75\right)^\circ$(−75)°?
Most of the time in this topic, you will evaluate trigonometric functions for given angles or find the angle that makes a ratio true. The following style of question asks you to find one trigonometric function having been given another. Perhaps counterintuitively, we do not need to find the actual angle involved at any stage! We simply use our understanding of the unit circle to ensure our ratio is presented with the correct sign. We use Pythagoras' theorem to find the exact length of any missing sides in a related triangle. These sides are used as the values in the trigonometric ratio requested.
If $\cos\theta=\frac{2}{5}$cosθ=25 and $\sin\theta<0$sinθ<0 find $\tan\theta$tanθ.
Think: If the cos ratio is positive but the sin ratio is negative, this angle $\theta$θ must lie in the fourth quadrant. The image below shows us the scenario:
Do: To evaluate $\tan\theta$tanθ, we must find the missing side in the triangle. Using Pythagoras, this side is:
$\sqrt{5^22^2}=\sqrt{21}$√52−22=√21
Remembering that $\tan\theta$tanθ is negative in the fourth quadrant, our final answer is:
$\tan\theta=\frac{\sqrt{21}}{2}$tanθ=−√212
If $\theta$θ is an angle such that $\sin\theta$sinθ$>$>$0$0 and $\cos\theta$cosθ$<$<$0$0, which quadrant(s) does it lie in?
quadrant $I$I
quadrant $II$II
quadrant $IV$IV
quadrant $III$III
quadrant $I$I
quadrant $II$II
quadrant $IV$IV
quadrant $III$III
Given that $\cos\theta=\frac{6}{7}$cosθ=−67 and $\tan\theta<0$tanθ<0, find $\sin\theta$sinθ.
Know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent).