Trigonometry/Geometric Definitions of Trig Functions

Geometrically defining tangent

In the previous section, we algebraically defined tangent as $\tan \theta =\frac{\sin \theta }{\cos \theta }$, 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 unit circle: (i.e. $x=1$). 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 (Q) on the drawn tangent. The ordinate (QP) 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, positive or negative, can be the input to sine or cosine — the result will be as if the largest multiple of 2π (or 360°) were subtracted from or added to 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|Right Angle Trigonometry]]
Previous Page: [[../The_unit_circle|The unit circle]]

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