Transcript reference angle

SECTION 14-4
• Right Triangles and Function Values
Slide 14-4-1
RIGHT TRIANGLES AND FUNCTION
VALUES
•
•
•
•
Right Triangle Side Ratios
Cofunction Identities
Trigonometric Function Values of Special Angles
Reference Angles
Slide 14-4-2
RIGHT TRIANGLE SIDE RATIOS
The next slide shows an acute angle A in standard
position. The definitions of the trigonometric
function values of angle A require x, y, and r. x and y
are the lengths of the two legs of the right triangle
ABC, and r is the length of the hypotenuse. The
functions of trigonometry can be adapted to describe
the ratios of these sides.
Slide 14-4-3
RIGHT TRIANGLE SIDE RATIOS
The side of length y is called the side opposite
angle A, and the side of length x is called the side
adjacent to angle A.
y
B
r
(x, y)
y
A
x
C
x
Slide 14-4-4
RIGHT TRIANGLE SIDE RATIOS
The lengths of these sides can be used to replace
x and y in the definitions of the trigonometric
functions, with r replaced by the length of the
hypotenuse.
Slide 14-4-5
RIGHT TRIANGLE-BASED DEFINITIONS
OF TRIGONOMETRIC FUNCTIONS
For any acute angle A in standard position,
y side opposite A
sin   
r
hypotenuse
x side adjacent to A
cos   
r
hypotenuse
y
side opposite A
tan   
x side adjacent to A
more
Slide 14-4-6
RIGHT TRIANGLE-BASED DEFINITIONS
OF TRIGONOMETRIC FUNCTIONS
r
hypotenuse
csc   
y side opposite A
r
hypotenuse
sec  
x side adjacent to A
x side adjacent to A
cot   
.
y
side opposite A
Slide 14-4-7
EXAMPLE: FINDING TRIGONOMETRIC
FUNCTION VALUES OF AN ACUTE
ANGLE
Find the values of the trigonometric functions for
angle A in the right triangle.
B
13
5
Solution
5
sin A 
13
13
csc A 
5
C
12
cos A 
13
13
sec A 
12
12
A
5
tan A 
12
12
cot A 
5
Slide 14-4-8
COFUNCTION IDENTITIES
a
sin A   cos B
c
c
sec A   csc B
b
a
tan A   cot B
b
B
c
a
C
b
A
Slide 14-4-9
COFUNCTION IDENTITIES
Because C = 90°, A and B are complementary
angles. Because A and B are complementary angles
and sin A = cos B, the functions sine and cosine are
called cofunctions. Also, tangent and cotangent are
cofunctions, as are secant and cosecant. And because
A and B are complementary angles, we have
B = 90° – A. This leads to
sin A = cos B = cos(90° – A).
The rest of the cofunction identities are on the next
slide.
Slide 14-4-10
COFUNCTION IDENTITIES
For any acute angle A,
sin A  cos(90  A)
csc A  sec(90  A)
cos A  sin(90  A)
sec A  csc(90  A)
tan A  cot(90  A)
cot A  tan(90  A)
Slide 14-4-11
EXAMPLE: WRITING FUNCTIONS IN
TERMS OF COFUNCTIONS
Write each of the following in terms of cofunctions.
a) cos 48°
b) tan 33°
c) sec 81°
Solution
a) sin 42°
b) cot 57°
c) csc 9°
Slide 14-4-12
TRIGONOMETRIC FUNCTION VALUES
OF SPECIAL ANGLES
Certain special angles, such as 30°, 45°, and 60°,
occur so often in applications of trigonometry
that they deserve special study. The exact
trigonometric function values of these angles,
found by the properties of geometry and the
Pythagorean theorem, are summarized on the
next slide.
Slide 14-4-13
TRIGONOMETRIC FUNCTION VALUES
OF SPECIAL ANGLES

sin 
cos
30°
1
2
3
2
3
3
3
45°
2
2
1
1
60°
3
2
2
2
1
2
3
3
3
tan 
cot 
sec
csc
2 3
3
2
2
2
2
2 3
3
Slide 14-4-14
REFERENCE ANGLES
Associated with every nonquadrantal angle in
standard position is a positive acute angle called its
reference angle. A reference angle for an angle  ,
written  , is the positive acute angle made by the
terminal side of angle  and the x-axis.
y

y


x

x
Slide 14-4-15
EXAMPLE: REFERENCE ANGLES
Find the reference angle for 232°
Solution
   232 180  52
y
232°
52°
x
Slide 14-4-16
EXAMPLE: REFERENCE ANGLES
Find the reference angle for 1020°
Solution
Find a coterminal angle between 0° and 360°:
1020° – 2(360°) = 300°
y
   360  300  60
300°
60° x
Slide 14-4-17
REFERENCE ANGLES, WHERE
Q II
0    360
   180  
 
y
y


QI

x
Q III
x
    180

   360  
Q IV
y
y

x

x
Slide 14-4-18
FINDING TRIGONOMETRIC FUNCTION
VALUES FOR ANY NONQUADRANTAL
ANGLE
Step 1 If  > 360°, or if  < 0°, find a
coterminal angle by adding or subtracting
360° as many times as needed to obtain an
angle greater than 0° but less than 360°.
Step 2
Find the reference angle  .
Step 3
Find the necessary values of the
trigonometric functions for the reference
angle  .
Slide 14-4-19
FINDING TRIGONOMETRIC FUNCTION
VALUES FOR ANY NONQUADRANTAL
ANGLE
Step 4
Determine the correct signs for the values
found in step 3. This result gives the
values of the trigonometric functions for
angle  .
Slide 14-4-20
EXAMPLE: FINDING TRIGONOMETRIC
FUNCTION VALUES USING A REFERENCE
ANGLE
Use a reference angle to find the exact value
of cos 495°.
Solution
Find a coterminal angle between 0° and
360°: 495° – 360° = 135°
   180 135  45
In quadrant II, so cosine
is negative.
2
cos 495  cos135   cos 45  
2
Slide 14-4-21