Download MTH113 Section 4.2

Section 4.2 (The Unit Circle)
The unit circle is the circle with radius 1 and center at the origin of the rectangular coordinate system. The equation
of the unit circle is given by
______________________
Arc length = s = rθ = 1(t) = t (radian measure of the central angle is the
same as the length of the intercepted arc)
t
One trigonometric function is defined as sin t = y (pronounced sine of t)
This is a function of t and you can see that the input to the function is t
(the angle in radians) and the output involves y (the y-coordinate of the
point on the unit circle that corresponds to angle t
t
(1,0)
Trigonometric functions defined…
csc t = 1 / y, y  0
sec t = 1 / x, x  0
cot t = x / y, y  0
sin t = y
cos t = x
tan t = sin t / cos t = y / x , x  0
 3 1
, 
 2 2


Example: Find the values of the 6 trigonometric functions for the angle t corresponding to the point P  
 3 1
P 
, 
 2 2


t
sin t =
csc t =
cos t =
sec t =
tan t =
cot t =
(1,0)
Example: Find the same values at P = (0 , –1)
We can use these formulas to find the values of trigonometric functions at t =  / 4, remembering that the formula
for our unit circle is x2 + y2 = 1 and knowing that the point at this angle has x = y (sketch and work on board)
The point P = (a , b) on the unit circle corresponding to t =  / 4 is (
Example: Use this point to find sin  / 4, csc  / 4, and cos  / 4
Consider whether sin t and cos t are even or odd functions by examining sin(– t) and cos(– t)
sin t = y , sin(– t) =
cos t = x , cos(– t) =
t
Cosine and secant functions are _____________ (cos(– t) = cos t, sec(– t) =sec t)
–t
Sine, cosecant, tangent, cotangent are _________ (sin(– t) = – sin t, tan(– t) = – tan t)
(1,0)
Example: Find the value of each of the following trigonometric functions
 
sec    
 4
 
sin    
 4
Since sin t = y and csc t = 1 / y, we also have csc t = 1 / sin t (we can use similar techniques to find the following)…
Reciprocal Identities
1
csc(t )
1
cos ( t ) 
se c( t )
1
tan ( t ) 
cot( t )
sin ( t ) 
Example: Given sin ( t ) 
Quotient Identities
1
sin ( t )
tan ( t ) 
1
cos ( t )
1
cot ( t ) 
tan ( t )
sin ( t )
cos ( t )
cot ( t ) 
cos ( t )
sin ( t )
csc( t ) 
sec( t ) 
2
5
and cos ( t ) 
, find the value of the remaining 4 trig functions…
3
3
We can explore other relationships among the trig identities by examining the equation of the unit circle
x2 + y2 = 1
=>
(cos t)2 + (sin t)2 = 1
Pythagorean Identities: sin 2 t + cos 2 t = 1
Example: Given that sin ( t ) 
=>
sin2 t + cos2 t = 1
1 + tan 2 t = sec 2 t
(common form)
1 + cot 2 t = csc 2 t
1

and 0  t  , find the value of cos t using a trig identity
2
2
Certain patterns in nature repeat again and again over a given period of time (tides every 12 hours for example).
Trigonometric functions are often used to model periodic things. Consider a point (x , y) on the unit circle that
travels along the perimeter over an angle of 2  radians (draw on board).
For any integer n and real number (angle) t
=>
sin(t + 2n) = sin t and cos(t + 2n) = cos t
sin, cos, csc, and sec have period 2. What about tan and cot (consider the same example and look at x and y
coordinates after travelling over an angle of  radians).
For any integer n and real number (angle) t
=>
tan(t + n) = tan t
tan has period 
 5 
 9 
 and cos  

 4 
 4 
Example: Find the value of cot 
Confirm that you can use a calculator to evaluate trigonometric functions (sin  / 4 and csc 1.5)