TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
8/7/2015
Section 7.4
Trigonometric Functions of General Angles
TRIGONOMETRIC FUNCTIONS
OF ANY ANGLE
Let be any angle in standard position, and let , denote
the coordinates of any point, except the origin 0, 0 , on the
terminal side of . If
denotes the distance from
0, 0 to , then the six trigonometric functions of are
defined as the ratios:
sin
cos
tan
csc
sec
cot
provided no denominator equals 0. If a denominator equals 0, that trigonometric function of the angle is not defined.
TRIGONOMETRIC FUNCTIONS OF QUADRANTAL ANGLES
0?; 0 90?; /2 180?; 270?; 3/2
sin 0 1 0 -1
cos 1
tan 0
csc
not defined
sec 1
cot
not defined
0
not defined
1
not defined
0
-1
0
not defined
-1
not defined
0
not defined
-1
not defined
0
COTERMINAL ANGLES
Two angles in standard position are said to be coterminal if they have the same terminal side
NOTE: Coterminal angles are NOT equal, they merely stop at the same place.
COTERMINAL ANGLES AND TRIGONOMETRIC FUNCTIONS
Because coterminal angles have the same terminal side, the values of the six trigonometric functions of coterminal angles are equal.
SIGNS OF THE TRIGONOMETRIC
FUNCTIONS
Sign of
sin csc cos sec tan cot
Terminal Side in Quadrant
I
II
III
IV
positive positive negative negative
positive negative negative positive
positive negative positive negative
1
8/7/2015
REFERENCE ANGLES
Let denote an angle that lies in a quadrant. The acute angle formed by the terminal side of and either the positive -axis or the negative -axis is called the reference angle for .
THE REFERENCE ANGLE
THEOREM
Reference Angle Theorem: If is an angle, in standard position, that lies in a quadrant and is its reference angle, then
sin
sin cos
cos tan
tan
csc
csc sec
sec cot
cot
where the + or - sign depends on the quadrant in which lies.
FINDING THE VALUES OF THE
TRIGONOMETRIC FUNCTIONS OF ANY
ANGLE
? If the angle is a quadrantal angle, draw the angle, pick a point on its terminal side, and apply the definition of the trigonometric functions.
? If the angle lies in a quadrant:
1. Find the reference angle of .
2. Find the value of the trigonometric function at .
3. Adjust the sign (+ or -) of the value of the trigonometric functions based on the quadrant in which lies.
2
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