Download 1 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS

1 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS
5.2 TRIGONOMETRIC FUNCTIONS
Topics :
• Trigonometric functions
• Quadrantal angles
• Reciprocal Identities
• Signs and ranges of function values
• Pythagorean identities
• quotient identities
TRIGONOMETRIC FUNCTIONS
To define the six trigonometric functions, we start with an angle θ (theta) in standard
position, and choose any point P having coordinates (x,y) on the terminal side of the
angle θ . We find the distance, r, from P(x,y) to the origin, (0,0) using the distance
formula r = x 2 + y 2 .
The radius is always positive.
The six trigonometric function of the angle θ are sine, cosine, tangent, cotangent, secant,
and cosecant. In the following definitions, we use the customary abbreviations for the
names of these functions:
Homework Exercises:
Sketch an angle θ ,theta, in standard position, such that θ has the smallest possible
positive measure, and the given point is on the terminal side of θ ,theta
2. ( −12, −5 )
Find the values of the 6 trig functions for each angle in standard position having the given
point on its terminal side. Rationalize denominators when applicable.
4. ( −4, −3)
6. (-4,0)
10. (3,-4)
12. If the terminal side of an angle θ is in QIII, what is the sign of each of the six trig
functions of θ ?
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Homework exercises 13 – 16. Suppose that the point (x,y) is in the indicated quadrant.
Decide whether the given ratio is positive or negative (draw a sketch)
y
14. III,
r
x
16. IV,
y
Homework exercises 17 -20.
An equation of the terminal side of an angle theta in standard position is given with a
restriction on x. Sketch the smallest positive such angle theta, and find the values of the
6 trig functions of theta.
18. 3x + 5 y = 0; x ≥ 0
20. −5 x − 3 y = 0; x ≤ 0
Quadrantal Angles
If the terminal side of an angle in standard position lies along the y-axis, any point on this
terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has
y-coordinate 0 for any point on the terminal side. Since the values of x and y appear in
the denominators of some trigonometric function, and since a fraction is undefined if its
denominator is 0, some trig function values of quadradantal angles are undefined.
Homework exercises
9. Find the 6 trig functions (-2,0)
21. Find the 6 trig function values of the quadrantal angle 450°
Undefined function Values
If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant
functions are undefined. If it lies along the x-axis, then the cotangent and cosecant
functions are undefined.
See table for quadrantal angles pp 485
Reciprocal Identities:
Identities are equations that are true for all values of the variable for which the
expressions are defined
Note: Identities can be written in different forms. For example,
1
1
sin θ =
csc θ =
( sin θ )( csc θ ) = 1
cos θ
sin θ
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Homework exercises 44-49. Use the appropriate reciprocal identity to find each function
value. Rationalize denominators when applicable.
44.
46.
48.
Signs and Ranges of Function Values
In the definitions of the trigonometric functions, r is the distance from the origin to the
point (x,y). Distance is never negative, so r > 0. If we choose a point (x,y) in quadrant I,
then both x and y will be positive. Thus, the values of all 6 functions will be positive in
quadrant I.
A point (x,y) in quadrant II has x <0 and y >0. This makes the values of sine and
cosecant positive for quadrant II angles, while the other four functions take on negative
values. Similar results can be obtained for the other quadrants, as summarized on page
487:
sin θ
cos θ
tan θ
cot θ
sec θ
csc θ
θ in Quadrant
I
+
+
+
+
+
+
II
+
+
III
+
+
IV
+
+
-
Homework exercises 57 – 60. Identify the quadrant or quadrants for the angle satisfying
the given conditions.
58.
cos θ > 0, tan θ > 0
60. tan θ < 0, cot θ < 0
Homework exercises 61 – 66
Give the signs of the sine, cosine, and tangent functions for each angle:
62. 183 degrees
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64. 412 degrees
66 -121 degrees
Ranges of the Trigonometric Functions
For any angle θ for which the indicated functions exists:
1. −1 ≤ sin θ ≤ 1 and − 1 ≤ cos θ ≤ 1
2. tan θ and cot θ can be any real number;
3. sec θ ≤ −1 or secθ ≥ 1 and cscθ ≤ -1 or cscθ ≥ 1
(notice that sec θ and cscθ are never between -1 and 1)
Homework exercises:70 -77
Decide whether each statement is possible or impossible for an angle θ .
70. sin θ = 2
72. tan θ = 0.92
74. sec θ = 1
1
76. sin θ = and cscθ =2
2
Pythagorean Identities:
sin 2 θ + cos 2 θ = 1
tan 2θ + 1 = sec2 θ 1+cot 2θ = csc2 θ
Quotient Identities:
sin θ
= tan θ
cos θ
cosθ
= cot θ
sinθ
Homework exercises 78 – 86. Use identities to find each function value. Use a
calculator in Exercises 84,85.
78.
80.
82.
84.
Homework exercises 87 – 92. Find all trigonometric function values for each angle.
Use a calculator in exercises 91 and 92.
88.
90.
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92.
Note: do some word problems 97-100 pp 493-494
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