Download reciprocal property of trigonometric functions.

WARM UP
• For θ = 2812° find a coterminal angle between 0°
and 360°.
292°
• What is a periodic function?
A function with repeating values
• What are the six trigonometric functions?
Sine, cosine, tangent, cotangent, secant, cosecant
Values of the Six
Trigonometric
Functions
OBJECTIVES
• In this section you will define four other trigonometric
functions
• Be able to find values of the six trigonometric functions
approximately, by calculator, for any angle and exactly for
certain special angles.
KEY TERMS & CONCEPTS
• Tangent
• Reciprocal properties
• Cotangent
• Unit circle
• Secant
• Ellipsis format
• Cosecant
• Complementary angles
THE 6 TRIGONOMETRIC
FUNCTIONS
• Sine and cosine have been defined for any angle as ratios of
the coordinates (u, v) of a point on the terminal side of the
angle and, equivalently, as ratios of the displacements in the
reference angle.
v vert.displacement
opposite
sin  

r
radius
hypotenuse
u hori.displacement
adjacent
cos  

r
radius
hypotenuse
• Four other ratios can be made using u, v and r. Their names
are tangent, cotangent, secant, and cosecant.
THE 6 TRIGONOMETRIC
FUNCTIONS
• The right triangle definition of tangent for an acute angle is
extended to the ratio of v to u for a point on the terminal side
of any angle.
v vert.displacement opposite
tan  

u hori.displacement adjacent
• The cotangent, secant and cosecant functions are reciprocal
of the tangent, cosine, and sine functions, respectively. The
relationship between each pair of functions, such as
cotangent and tangent, is called the reciprocal property of
trigonometric functions.
THE 6 TRIGONOMETRIC
FUNCTIONS
• When you write the functions in a column in the order sin θ,
cos θ, tan θ, cot θ, sec θ, csc θ, the functions and their
reciprocals have this pattern.
sin θ
cos θ
tan θ
cot θ
sec θ
csc θ
reciprocals
1
cot  
tan 
DEFINITIONS
• Let (u, v) be a point r units from the origin on the terminal side of a rotating
ray. If θ is the angle to the ray, in standard position, then the following
definitions hold.
Right Triangle Form
Coordinate Form
Note: The
coordinates u and v
are also the
horizontal and
vertical
displacements of the
point (u, v) in the
reference triangle.
THE NAMES TANGENT
AND SECANT
• To see why the names tangent and secant are used, look at the
graph. The point (u, v) has been chosen on the terminal side
of angle θ where r = 1 unit. The circle traced by (u, v) is called
the unit circle (a circle with radius 1 unit). The value of sine is
given by
v v
sin     v
r 1
TANGENT
• Thus the sine of an angle is equal to the vertical
coordinate of a point on the unit circle. Similarly,
cos θ = u/1 = u, the horizontal coordinate of a point
on the unit circle.
opposite length tan segment
tan 

 length tan segment
adjacent
1
• Hence the name tangent is used.
SECANT
• The hypotenuse of this larger reference triangle is
part of a secant line, a line that cuts through the
circle. In the larger triangle,
hypotenuse lengthsec segment
sec 

 lengthsec segment
adjacent
1
• Hence the name secant is used. By properties of
similar triangles, for any point (u, v) on the terminal
v
r
side tan   
sec  
and
u
u
EXAMPLE 1
• Example 1 shows how you can find approximate values of
all six trigonometric functions using your calculator.
• Evaluate by calculator the six trigonometric functions of
58.6°. Round to four decimal places.
Solution: You can find sine, cosine, and tangent directly on the
calculator.
sin 58.6° = 0.8535507. . . ≈ 0.8536
cos 58.6° = 0.5210096. . . ≈ 0.5210
tan 58.6° = 1.6382629. . . ≈ 1.6383
Note preferred usage of the ellipsis
and the ≈ sign.
EXAMPLE 1 CONT.
• The other three functions are the reciprocals of the sine,
cosine, and tangent functions. Notice that the reciprocals
follow the pattern described earlier.
1
cot 58.6 
 0.61040260...  0.6104
tan 58.6
1
sec 58.6 
 1.91935031...  1.9194
cos 58.6
1
csc 58.6 
 1.17157642...  1.1716
sin 58.6
EXACT VALUES BY
GEOMETRY
•
If you know a point on the terminal side of an angle, you
can calculate the values of the trigonometric functions
exactly. Example 2 shows you the step
• The terminal side of angle θ contains the point (−5, 2). Find
exact values of the six trigonometric functions of θ. Use
radicals if necessary, but no decimals.
EXAMPLE 2
Solution: You can find sine, cosine, and tangent directly on the
calculator.
•
Sketch the angle in standard position
• Pick a point on the terminal side, (-5, 2) in this instance and
draw a perpendicular.
• Mark the displacements on the reference triangle, using the
Pythagorean theorem.
EXAMPLE 2 CONT.
r  (5)  2  29
2
vertical 2
sin 

radius
29
horizontal 5
5
cos 


radius
29
29
vertical
2
2
tan 
 
horizontal 5 5
2
1
5
cot  

tan  2
1
29
sec 

cos
5
1
29
csc 

sin 
2
EXAMPLE 2 CONT.
• Note: You can use the proportions of the side
lengths 30°-60°-90° triangle and the 45°-45°-90°
triangle to find exact function values for angles
whose reference angle measure is a multiple of 30°
or 45°. The graphs show these proportions.
EXAMPLE 3
Find the exact values (no
decimals) of the six trigonometric
functions of 300°
SOLUTION: Sketch an angle terminating in
Quadrant IV and a reference triangle.
 3
3
sin  

2
2
Use the negative square root
because v is negative.
1
cos 
2
 3
tan 
 3
1
1
1
cot  

tan
3
Use
reciprocal
property
1
2
sec 
 2
cos 1
Simplify
1
2
2
csc 


sin  3
3
EXAMPLE 3 CONT.
Note: To avoid errors in placing the 1, 2, and √3 on
the reference triangle, remember that the
hypotenuse is the longest side of a right triangle
and that 2 is greater than √3.
EXAMPLE 4
This example will show you how to find the function values
for an angle that terminates on a quadrant boundary.
Example 4: Without using a calculator, evaluate the six
trigonometric functions for an angle of 180°.
Do the obvious
simplification
SOLUTION
The graph shows a 180° angle in standard position.
The terminal side falls on the negative side of the
horizontal axis. Pick any point on the terminal side,
such as (-3, 0). Note that although the u-coordinate
of the point is negative, the distance r from the origin
to the point is positive because it is the radius of a
circle. The vertical coordinate, v, is 0.
vertical 0
sin180 
 0
radius 3
Use “vertical, radius” rather
than “opposite, hypotenuse.”
horizontal 3
cos180 
  1
radius
3
vertical 0
tan180 
 0
horizontal 3
Do the obvious
simplification
1
1
cot180 
  undefined
tan180 0
1
1
sec180 
  1
cos180 1
1
1
csc180 
  undefined
sin180 0
CH. 2.4 ASSIGNMENTS
• Textbook pg. 78 #2 through 40, every fourth one &
44.