Unit circle and trig functions (degrees)

 

In right-angled triangle trigonometry, we can only deal with angles whose sizes are between $0^\circ$0° and $90^\circ$90°. In the chapter on Angles of Any Magnitude, it is explained how the sine, cosine and tangent functions are given a more general definition so that they can be applied to angles that are impossible in right-angled triangle trigonometry. The functions are defined in terms of the coordinates of a point that is free to move on the unit circle, in the following way.

The radius, drawn to the point on the circle,  makes an angle with the positive horizontal axis. By convention, the angle is measured anticlockwise from the axis. It can have any real value, positive or negative.

We define the cosine of the angle to be the horizontal coordinate of the point, and we define the sine of the angle to be the vertical coordinate of the point. Then, the tangent of the angle $\alpha$α  is defined to be the fraction $\frac{\sin\alpha}{\cos\alpha}$sinαcosα​.

By considering the coordinates of a point as it moves around the unit circle, we see that the sine and cosine functions have either a positive or a negative sign depending on which quadrant the angle is in. The signs depend on the signs of the coordinates involved. 

Have a look at the following applet, you can make settings for degrees or radians, and whether you want to rotate postiviely (anticlockwise) or negatively (clockwise).  As you explore, see if you can determine where sine, cosine and tangent are positive or negative.

 

The sine function is positive in the first and second quadrants and negative in the others while the cosine function is positive in the first and fourth quadrants. Consequently, the tangent function is positive 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.

 

We use the unit circle, also, to develop a more natural way of measuring angles. From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. The angle subtended at the centre by any one of those arcs, is called one degree. In effect, an arc of the circle is used as a measure of the angle it subtends.

In a similar way, we now restrict our attention to circles of radius one unit, and we measure angles subtended at the centre by arcs of this circle. This method of measuring angles is called radian measure.

We know that the circumference of the unit circle is $2\pi$2π, so the angle represented by a full turn around the circle is $2\pi$2π radians. This is equivalent to $360^\circ$360°. A half-circle makes an angle of $\pi$π radians or $180^\circ$180° while a right-angle is $\frac{\pi}{2}$π2​ radians. 

An angle of $1$1 radian must be $\frac{360^\circ}{2\pi}\approx57.3^\circ$360°2π​≈57.3° .

In practice, angles given in radian measure are usually expressed as fractions of $\pi$π. 

Because angles in radian measure are in essence just fractions of the unit circle, they do not require a unit, although some writers indicate that radian measure is being used by adding a superscript c after a number, like this: $\frac{\pi}{6}^{^c}$π6​c . (The c is short for circular-measure.)

Trigonometric functions of a variable angle measured in radians can thus be thought of, like other functions, as functions of a  real number. Thus, we encounter function definitions like $y=\sin x$y=sinx, where $x$x is understood to be an ordinary number.

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

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

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°

The angle $28^\circ$28° is equivalent to $\frac{28}{360}$28360​ of the full circle. So, in radian measure it is $\frac{28}{360}\times2\pi=\frac{7\pi}{45}$28360​×2π=7π45​. So, the function values are:

$\sin\frac{7\pi}{45}$sin7π45​, $-\cos\frac{7\pi}{45}$−cos7π45​ and $-\tan\frac{7\pi}{45}$−tan7π45​.

More Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3