How do you find values of trigonometric functions using the unit circle?

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How do you find values of trigonometric functions using the unit circle?

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Answer 1 · 2015-04-22T15:41:13

Given an arc $A M = x$ $AM = x$ , with origin $A$ $A$ and extremity $M$ $M$ , that rotates on the trig unit circle with origin $O$ $O$ .
The value of $cos x$ $cos x$ is given by the projection of $M$ $M$ on the horizontal $O A x$ $OAx$ . $O m = cos x$ $Om = cos x$ .
The value of $sin x$ $sin x$ is given by the projection of $M$ $M$ on the vertical $O B y$ $OBy$ axis. $O n = sin x$ $On = sin x$ . $B$ $B$ is the top point of the trig circle.
Prolong the radius $O M$ $OM$ until it meets the vertical axis $A T$ $AT$ at $t$ $t$ . the segment $A t = tan x$ $At = tan x$ .
Prolong the radius $O M$ $OM$ until it meets the horizontal $B Z$ $BZ$ at $z$ $z$ . The segment $B z = cot x$ $Bz = cot x$ .
In summary, the trig unit circle defines 4 trig functions of the arc $A M = x$ $AM = x$ . When the arc extremity $M$ $M$ rotates, each function: $f (x) = cos x$ $f(x) = cos x$ ; $f (x) = sin x$ $f(x) = sin x$ ; $f (x) = tan x$ $f(x) = tan x$ ; and $f (x) = cot x$ $f(x) = cot x$ varies along its own axis.

For example, the function $f (x) = sin x$ $f(x) = sin x$ varies from $1$ $1$ to $- 1$ $-1$ then back to $1$ $1$ on the horizontal $O A x$ $OAx$ axis.
For example, the function $f (x) = tan x$ $f(x) = tan x$ varies from $0$ $0$ to $+ \infty$ $+infty$ on the vertical $A T$ $AT$ axis, when $x$ $x$ varies from $0$ $0$ to $\frac{π}{2}$ $pi/2$ . And $f (x) = tan x$ $f(x) = tan x$ varies from $- \infty$ $-infty$ to $0$ $0$ when $x$ $x$ moves from $\frac{π}{2}$ $pi/2$ to $π$ $pi$ .

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How do you find values of trigonometric functions using the unit circle?

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