Properties of Trigonometric Functions
The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x)are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points. Sine Function: f(x) = sin (x)

* Graph

* Domain: all real numbers
* Range: [-1 , 1]
* Period = 2pi
* x-intercepts: x = k pi , where k is an integer.
* y-intercepts: y = 0
* Maximum points: (pi/2 + 2 k pi , 1) , where k is an integer. * Minimum points: (3pi/2 + 2 k pi , -1) , where k is an integer. * Symmetry: since sin(-x) = - sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0 , 0). * Intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0, pi/2) and (3pi/2 , 2pi), and decreasing on the interval (pi/2 , 3pi/2). Cosine Function : f(x) = cos (x)

* Graph

* Domain: all real numbers
* Range: [-1 , 1]
* Period = 2pi
* x intercepts: x = pi/2 + k pi , where k is an integer. * y intercepts: y = 1
* maximum points: (2 k pi , 1) , where k is an integer.
* minimum points: (pi + 2 k pi , -1) , where k is an integer. * symmetry: since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. * intervals of increase/decrease: over one period and from 0 to 2pi, cos (x) is decreasing on (0 , pi) increasing on (pi , 2pi).

Tangent Function : f(x) = tan (x)
* Graph

* Domain: all real numbers except pi/2 + k pi, k is an integer. * Range: all real numbers
* Period = pi
* x intercepts: x = k pi , where k is an integer.
* y intercepts: y = 0
* symmetry: since tan(-x) = - tan(x) then tan (x) is an odd function and its graph is symmetric with respect the origin. * intervals of increase/decrease: over one period and from -pi/2 to pi/2, tan (x) is increasing....

...Section 5.2 TrigonometricFunctions of Real Numbers
The TrigonometricFunctions
EXAMPLE: Use the Table below to ﬁnd the sixtrigonometricfunctions of each given real number t. π π (a) t = (b) t = 3 2
1
EXAMPLE: Use the Table below to ﬁnd the sixtrigonometricfunctions of each given real number t. π π (a) t = (b) t = 3 2 Solution: (a) From the Table, we see that the terminal point determined by √ t = √ is P (1/2, 3/2). Since the coordinates are x = 1/2 and π/3 y = 3/2, we have √ √ π 3 3/2 √ π 1 π sin = cos = tan = = 3 3 2 3 2 3 1/2 √ √ π 3 2 3 π π 1/2 csc = = sec = 2 cot = √ 3 3 3 3 3 3/2 (b) The terminal point determined by π/2 is P (0, 1). So π π 1 π 0 π cos = 0 csc = = 1 cot = = 0 sin = 1 2 2 2 1 2 1 But tan π/2 and sec π/2 are undeﬁned because x = 0 appears in the denominator in each of their deﬁnitions. π . 4 Solution: √ From the Table above, we see that √ terminal point determined by t = π/4 is the √ √ P ( 2/2, 2/2). Since the coordinates are x = 2/2 and y = 2/2, we have √ √ √ π 2 2 2/2 π π sin = =1 cos = tan = √ 4 2 4 2 4 2/2 √ π √ π π √ 2/2 csc = 2 sec = 2 cot = √ =1 4 4 4 2/2 EXAMPLE: Find the sixtrigonometricfunctions of each given real number t =
2
Values of the TrigonometricFunctions
EXAMPLE: π π (a) cos > 0, because the terminal...

...aptitude for math improves critical thinking and promotes problem-solving abilities. One specific area of mathematical and geometrical reasoning is trigonometry which studies the properties of triangles. In some of the fields such as architecture, astronomy , biology, cartography, chemistry, civil engineering, computer graphics, geophysics, crystallography and economics , some things in these fields cannot be understood without trigonometry. These proved that trigonometry really is needed in our real life situation.
Trigonometry finds a perfect partner in modern architecture. It is really needed in architecture fields, without trigonometry architecture is hard to understand. The beautifully curved surfaces in steel, stone and glass would be impossible if not for the immense potential of this science. So how does this work actually. In fact the flat panels and straight planes in the building are but at an angle to one another and the illusion is that of a curved surface.
Although it is unlikely that one will ever need to directly apply a trigonometricfunction in solving a practical issue, the fundamental background of the science finds usage in an area which is passion for many is music. As we may be aware sound travels in waves and this pattern though not as regular as a sine or cosine function, is still useful in developing computer music. A computer cannot obviously listen to and comprehend music as we do, so...

...
PROPERTIES OF SINE AND COSINE FUNCTIONS:
1. The sine and cosine functions are both periodic with period 2π.
2. The sine function is odd function since it’s graph is symmetric with respect to the origin, while the cosine function is an even function since it’s graph is symmetric with respect to y axis.
3. The sine functions:
a. Increasing in the intervals[0, π/2]and [3π/2, 2π]; and
b. Decreasing in the interval [π/2, 3π/2],over a period of 2 π.
4. The cosine function is:
a. Increasing in the interval [π, 2π]; and
b. Decreasing in the interval [0, π], over a period 2π.
5. Both the sine and cosine functions are continuous functions.
6. The domain of the sine and cosine functions is the set of all real numbers from -1 to 1
7. The amplitude of both the sine and cosine functions is 1, since one-half of the sum of the lower bound is 1, that is ½[|1|]+[|-1|]=2/2 or 1.
8. The maximum and minimum values of the sine and cosine functions are 1 and -1 respectively, which occur alternately midway between the points where the functions is zero.
SINE FUNCTION COSINE FUNCTION
QUADRANT
AS S VARIES
VALUES OF SIN S
VALUES...

...Trigonometric Identities
I. Pythagorean Identities
A. [pic]
B. [pic]
C. [pic]
II. Sum and Difference of Angles Identities
A. [pic]
B. [pic]
C. [pic]
D. [pic]
E. [pic]
F. [pic]
III. Double Angle Identities
A. [pic]
B. [pic]
=[pic]
=[pic]
C. [pic]
IV. Half Angle Identities
A. [pic]
B. [pic]
C. [pic]
6-1 Inverse Trig Functions p. 468: 1-31 odd
I. Inverse Trig Functions
A. [pic]
B. [pic]
C. [pic]
Find the exact value of each expression
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
Use a calculator to find each value.
7. [pic] 8. [pic] 9. [pic]
Find the exact value of each expression.
10. [pic] 11. [pic] 12. [pic]
6-2 Inverse Trig Functions Continued p. 474:1-41 odd
I. Inverse Trig Functions
A. [pic]
B. [pic]
C. [pic]
Find the exact value of each expression.
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic] 7. [pic]
Find the exact value of each.
8. [pic] 9. [pic] 10. [pic]
Use a calculator to find each value.
11. [pic] 12. [pic]
Trigonometric Identities Trig Identities Worksheet: 1-6 all, 9, 13, 15, 19
I. Reciprocal...

...The properties of the 6 trigonometricfunctions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) include the domain, range, period, asymptotes and amplitudes.
The domain of a cosine and sine function is all real numbers and the range is -1 to 1. The period is 2π, and the amplitude is 1. They have no asymptotes. The domain of tangent is all real numbers except for π2+kπ. The range is all real numbers and the period is π. Tan has no amplitude and has asymptotes when x= π2+kπ.
The domain of a secant function is all real numbers except for π2+kπ. The domain of a cosecant function is all real numbers except for kπ. The range of both is (-∞.-1]U[1,∞) and the period is 2π. Secant has asymptotes when x=π2+kπ. Cosecant has asymptotes when x=kπ. They have no amplitude. Cotangent’s domain is all real numbers except for kπ. The range is all real numbers and the period is π. It has no amplitude and has asymptotes when x=kπ.
In an inverse function, the x coordinate, or the domain, and the y coordinate, the range, switch places. Since only one to one functions have inverses, we take the interval -π2 to π2, which contains all the possible values of the sine function. Now, the new domain is [-π2, π2], while the range stays the same. We then switch the domain and the range, so the domain and range of arcsin (x) is [-1,1] and [-π2, π2]. For cosine, the...

...Ashley Washington
Final exam essay
6/2/14
SixTrigonometricfunctions
The sixtrigonometricfunctions are found in a right triangle because they contain a right angle. The measures of the three angles, labeled A, B, and C. we used lower case letters a, b, and c to denote the lengths of the sides opposite of angles A, B, and C respectively. These sixtrigonometricfunctions are sine, cosine, and tangent, which are often used the most. The other three are cotangent, secant, and cosecant. However, it is said that in a right triangle the trigonometric ratios the sine, the cosine, and so on are functions of the acute angle. They depend only on the acute angle. For example, each value of sin theta represents the ratio of the opposite side to the hypotenuse, in every right triangle with that acute angle. If angle theta is 28 degrees, so then in every right triangle with a 28 degrees angle, its sides will be in the same ratio. We read it as, Sin. 28 degrees equals .469. This means that the right triangle have an acute angle of 28 degrees, which is half of the opposite angle. On the other hand, these sixtrigonometricfunctions have different labels names for each side. For example, the side label lower case “c” has a special name because it is the side opposite of the right angle...

...CERAE
CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
CONTENTS
-Angles and Their Measures
-Degrees and Radians
-Angles in Standard Position and Coterminal Angles
-Angles in a Quadrant
-The Unit Circle
-Coordinates of Points on the Unit Circle
-The Sine and Cosine Function
-Values of Sine and Cosine Functions
-Graphs of Sine and Cosine Functions
-The Tangent Function
-Graph of Tangent Function
-Trigonometric Identities
-Sum and Difference of Formulas for Sine and Cosine
-TrigonometricFunctions of an angle
-Values of the Function of an Angle
-Simple Trigonometric Equations
-Right Triangle Trigonometry
-Angle of Elevation/Depression
- Solving Right Triangles
-The Law of Sines
-The Law of Cosines
CERAE
CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
EXPERIENCE
What I Have Learned?
I learned many things in trigonometry especially in chapter 6 , I learned many lessons specifically in the lessons 6.1-6.4 (Angles and Their Measures, Degrees and Radians, Angles in Standard Position and Coterminal Angles, Angles in a Quadrant, The Unit Circle, Coordinates of Points on the Unit Circle, The Sine and Cosine Function , Values of Sine and Cosine Functions, Graphs of Sine and Cosine Functions, and The Tangent Function).
How Did I...

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