6.4 – Trigonometric Functions  Now, we will consider an acute angle  in standard position. And choose a point P with coordinates  x , y  on the terminal side, and draw a right triangle. The side adjacent to  has length x and the side opposite  has length y. The length of the hypotenuse, r, is the distance from the origin, which can be found by

x2  y2  r 2 , so r  x2  y2 .

y hypotenuse opposite

  Now, the trigonometric ratios can be written in terms of x, y, and r. y x y sin  cos  tan  r r x x r r cot  sec  csc  y x y

x adjacent

EXAMPLE 1: Find the sine, cosine and tangent of the angle  , whose terminal side passes through 2, 1 . (HINT: First, you need to find r.)

 Let t be a real number such that it is an angle of t in radians in standard position. If any point  x , y  is on y r r csc t  y sint 

the terminal side of angle t, then:

x r r sec t  x

cos t 

y x x cot t  y

tant 

where r  x2  y2 is the distance from  x , y  to the origin.

x2  y2  1 , so now sint 

 In the UNIT CIRCLE, the radius 

y y x x   y and cos t    x . r 1 r 1

 Now, we have new Unit Circle descriptions of Trigonometric Functions: P has coordinates  cos t , sin t  tan t 

y sin t  x cos t

cot t 

x cos t  y sin t

sec t 

1 1  x cos t

csc t 

1 1  y sin t

 NOW we can talk about domain and range of these functions:  The domain of the sine and cosine functions is the set of all real numbers  The range of the sine and cosine functions is the set of all real numbers between – 1 and 1 (inclusive), that is, the interval 1, 1 

Think about it, since the sine and cosine ratios have the radius of the circle in the denominator, the numerator will never be greater than the denominator!!! y  The tangent function is defined as tan t  , whenever x  0 , that is, for all points on the unit x  3 circle except at  0, 1 and  0, 1 , which is at and all angles coterminal to these. and 2 2  The domain of the tangent function is the set of all real numbers except: 



2

 2k , where k  0,  1,  2,  3,...

 The range of the tangent function is the set of all real numbers.  The next thing you need to pay attention to is the sign of each function in each quadrant, and of course, this is related to the sign of x and y in each quadrant: y Quadrantal Angles

Quad II



t 

 ,  

2 sint  cos t  tant 

Quad III

3  t  2 sint  cos t  tant 

 ,  

0t 



Quad I

 ,  

2 sint  cos t  tant 

3  t  2 2 sint  cos t  tant 

x

Quad IV

 ,  

EXAMPLE 2: Find the exact value of the trig functions at the quadrantal angles on the Unit Circle ( radius  1 ) T sin t

cos t tan t cot t

sec t csc t

0

 2



3 2

2

 For an angle  in standard position, the reference angle t ' or  ' is the positive acute angle formed by the terminal side of t or  and the x-axis.

t' t

(NOTE: These angles are NEVER made with the y-axis!!!!!)

t

t

t

t’ t’

t’

t'  t

t'    t

t'  t  

t '  2  t

 ' 

 '  180  

 '    180

 '  360  

 To find the sine, cosine or tangent of t radians: o Sketch the angle t in standard position and determine in which quadrant the terminal side falls. o Find the reference angle, t’. o Find the sine, cosine, or tangent of t’ and give it the appropriate sign. EXAMPLE 3: Use reference angles to find the exact values of sin t , cos t , and tan t . 5 5 5 a. t  b. t  c. t  4 3 6

 If an angle is larger than 2 , it is possible to find a coterminal angle between 0 and 2 , and then use its reference angle to find the trig values. EXAMPLE 4: Find the sine, cosine and tangent of

9 4

The COMPLETE Unit Circle

30  60  90

45  45  90

30  60  90