Trigonometry/Trigonometric Angular Functions

Geometrically defining sine and cosine

In the unit circle shown here, a unit-length radius has been drawn from the origin to a point (x,y) on the circle.

Defining sine and cosine

A line perpendicular to the x-axis, drawn through the point (x,y), intersects the x-axis at the point with the abscissa x. Similarly, a line perpendicular to the y-axis intersects the y-axis at the point with the ordinate y. The angle between the x-axis and the radius is $\alpha$.

We define the basic trigonometric functions of any angle $\alpha$ as follows:

Template:TrigBoxOpen $\begin{array}{cccc}\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{:}& \sin (\alpha )& =& y\\ \mathrm{C}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{:}& \cos (\alpha )& =& x\end{array}$

$\tan \theta$ can be algebraically defined.

$\tan \theta =\frac{\sin \theta }{\cos \theta }\cos \theta \ne 0$

$\tan \alpha =\frac{y}{x}x\ne 0$

These three trigonometric functions can be used whether the angle is measured in degrees or radians as long as it specified which, when calculating trigonometric functions from angles or vice versa.

Geometrically defining tangent

In the previous section, we algebraically defined tangent, and this is the definition that we will use most in the future. It can, however, be helpful to understand the tangent function from a geometric perspective.

Geometrically defining tangent

A line is drawn at a tangent to the circle: $x=1$. Another line is drawn from the point on the radius of the circle where the given angle falls, through the origin(O), to a point on the drawn tangent(Q). The ordinate of this point is called the tangent of the angle.

The slope of the line OQ = $\frac{KC}{OC}$ and as we mentioned before

KC = sin(θ) , OC = cos(θ)

Hence , the Slope of the line OQ = $\frac{\sin (\theta )}{\cos (\theta )}$

and also the slope of OQ = $\frac{QP}{OP}$ = $\frac{QP}{1}$ = $\frac{\tan (\theta )}{1}$ = tan(θ)

Hence , we can deduce that tan(θ) = $\frac{KC}{OC}$ = $\frac{\sin (\theta )}{\cos (\theta )}$ = QP = the ordinate of the point Q = the slope of OQ

Domain and range of circular functions

Any size angle can be the input to sine or cosine — the result will be as if the largest multiple of 2π (or 360°) were subtracted from the angle. The output of the two functions is limited by the absolute value of the radius of the unit circle, $|1|$.

Template:TrigBoxOpen $\begin{array}{ccc}& \mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}& \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\\ \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}& ℝ& [-1,1]\\ \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}& ℝ& [-1,1]\end{array}$

R represents the set of all real numbers. Template:TrigBoxClose

No such restrictions apply to the tangent, however, as can be seen in the diagram in the preceding section. The only restriction on the domain of tangent is that odd integer multiples of $\frac{\pi }{2}$ are undefined, as a line parallel to the tangent will never intersect it.

Template:TrigBoxOpen $\begin{array}{ccc}& \mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}& \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\\ \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}& ℝ\setminus \{\cdots ,-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},\cdots \}& ℝ\end{array}$ Template:TrigBoxClose

for a deep understanding of trigonometric functions explore this Applet

Applying the trigonometric functions to a right-angled triangle

If you redefine the variables as follows to correspond to the sides of a right triangle:
- x = a (adjacent)
- y = o (opposite)
- a = h (hypotenuse)

Next Page: Right Angle Trigonometry
Previous Page: The unit circle

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