SIX TRIGONOMETRIC FUNCTIONS

Topics in this lesson:
1. DEFINITION AND EXAMPLES OF THE TRIGONOMETRIC
FUNCTIONS OF AN ACUTE ANGLE IN TERMS OF A RIGHT
TRIANGLE
2. USING A RIGHT TRIANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS OF ANGLES IN THE FIRST, SECOND,
THIRD, AND FOURTH QUADRANTS
1. DEFINITION AND EXAMPLES OF THE TRIGONOMETRIC
FUNCTIONS OF AN ACUTE ANGLE IN TERMS OF A RIGHT
TRIANGLE
?
hypotenuse hypotenuse
opposite side adjacent side
of ? of ?
?
adjacent side opposite side
of ? of ?
Definition Given the angle ? in the triangle above. We define the following
hyp
adj
cos ? ?
adj
hyp
sec? ?
hyp
opp
sin ? ?
opp
hyp
csc? ?
adj
opp
tan ? ?
opp
adj
cot ? ?
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Illustration of the definition of the cosine function, sine function, tangent function,
secant function, cosecant function, and cotangent function for the acute angle ?
using right triangle trigonometry.
Second illustration of the cosine function, sine function, tangent function, secant
function, cosecant function, and cotangent function for the acute angle ? using
right triangle trigonometry.
Illustration of the definition of all the six trigonometric functions for an acute angle
? using right triangle trigonometry. Second illustration of all the six trigonometric
functions.
NOTE: Since the three angles of any triangle sum to 180 ? and the right angle in
the triangle is 90 ? , then the other two angles in the right triangle must sum to 90 ? .
Thus, the other two angles in the triangle must be greater than 0 ? and less than
90 ? . Thus, the other two angles in the triangle are acute angles. Thus, the angle ?
above is an acute angle. If we consider the angle ? in standard position, then ? is
in the first quadrant and we would have the following:
y
r
P ( ) ( x, y ) r ? ?
hypotenuse = r y = opposite side of ?
?
- r x =