 7.06 The unit circle

Lesson

If we restrict our attention to right-angled triangles, then we can only think of the trigonometric functions of angles whose size is between $0^\circ$0° and $90^\circ$90°. We redefine our definitions to make it possible to apply the functions to angles of any magnitude. Observe that the definitions given below are consistent with the original right-angled triangle definitions while being useful in a wider range of problems.

Consider the unit circle, radius $=$=$1$1unit, centred at the origin on the Cartesian plane. A radius to any point, $P$P on the circle, that 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 $y$y-coordinate of point $P$P.

The cosine of the angle is defined to be the $x$x-coordinate of point $P$P.

The tangent of the angle can be geometrically defined to be $y$y-coordinate 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 also define this algebraically as the ratio $\left(\frac{\sin\theta}{\cos\theta}\right)$(sinθcosθ). This also represents the gradient of the line that forms the angle $\theta$θ to the positive $x$x-axis.

Using the applet below, change the angle and take note of the sign (positive or negative) that each function has in the four different quadrants.

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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, 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 or by means of the mnemonic ASTC: 'All Stations To Central: All-Sine-Tangent-Cosine' that shows which functions are positive in each quadrant. These facts become important when trigonometric equations are being solved for all the solutions within a given range. From the applet we can also see that the value 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 can also use symmetry within the circle to find equivalent expressions. From the diagram we can see that the $y$y-coordinate of point $A$A is the same as point $B$B, hence: $\sin\theta=\sin\left(180^\circ-\theta\right)$sinθ=sin(180°θ).

We can also see the $x$x-coordinate of $A$A is the same as point $D$D, hence: $\cos\theta=\cos\left(360^\circ-\theta\right)$cosθ=cos(360°θ).

The $x$x-coordinate of $C$C has the same magnitude but would be negative in comparison to the $x$x-coordinate of $A$A, hence: $\sin\left(180^\circ+\theta\right)=-\sin\theta$sin(180°+θ)=sinθ.

What other equivalences can you see? What about angles larger than $360^\circ$360° or negative angles?

Worked examples

Example 1

Express $\cos117^\circ$cos117° in terms of a first quadrant angle.

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. Therefore, $\cos117^\circ$cos117° must be the same as $-\cos\left(180^\circ-117^\circ\right)=-\cos63^\circ$cos(180°117°)=cos63°. Example 2

Evaluate $\cos\left(120^\circ\right)$cos(120°) and find two equivalent expressions for this ratio.

To evaluate the ratio, enter it in your calculator, hence: $\cos120^\circ=\frac{-1}{2}$cos120°=12

The angle $120^\circ$120° is in the second quadrant. We could find an equivalent statement in the first quadrant by noting the $x$x-coordinate will have the same magnitude but opposite sign, so $\cos120^\circ$cos120° is equivalent to $-\cos\left(180^\circ-120^\circ\right)=-\cos60^\circ$cos(180°120°)=cos60°. We can find an equivalent statement in the third quadrant by noting the $x$x-coordinate would be the same, so $\cos120^\circ$cos120° is equivalent to $\cos\left(180^\circ+60^\circ\right)=\cos240^\circ$cos(180°+60°)=cos240°. We could find many more equivalent statements by adding or subtracting multiples of a full rotation( $360^\circ$360°), to the angle in our current expressions, hence $\cos480^\circ$cos480°$\cos\left(-600^\circ\right)$cos(600°) and $-\cos420^\circ$cos420° are all equivalent. We can double check this by evaluating each expression in the calculator, each should be $\frac{-1}{2}$12.

Example 3

Express the sine, cosine and tangent functions of the angle $512^\circ$512° in terms of an angle in the first quadrant.

The angle $512^\circ$512° is more than once around the full circle. So, it is equivalent to $512^\circ-360^\circ=152^\circ$512°360°=152°, which is in the second quadrant. We subtract the angle from $180^\circ$180° to find:

$\sin512^\circ=\sin28^\circ$sin512°=sin28°

$\cos512^\circ=-\cos28^\circ$cos512°=cos28°

$\tan512^\circ=-\tan28^\circ$tan512°=tan28°

Practice questions

Question 1

Which of the following will have positive answers? Select all correct answers.

1. $\tan296^\circ$tan296°

A

$\sin120^\circ$sin120°

B

$\cos91^\circ$cos91°

C

$\sin296^\circ$sin296°

D

$\cos120^\circ$cos120°

E

$\cos296^\circ$cos296°

F

$\tan296^\circ$tan296°

A

$\sin120^\circ$sin120°

B

$\cos91^\circ$cos91°

C

$\sin296^\circ$sin296°

D

$\cos120^\circ$cos120°

E

$\cos296^\circ$cos296°

F

Question 2

Evaluate $\cos126^\circ$cos126° correct to 2 decimal places and make note of the sign of your answer.

Question 3

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.

1. $\sin93^\circ$sin93°

2. $\cos195^\circ$cos195°

3. $\tan299^\circ$tan299°

Outcomes

4.2.1.2

understand the unit circle definition of 𝑐os(𝜃), 𝑠in(𝜃) and 𝑡an(𝜃) and periodicity using degrees and radians