Sine, Cosine, and Tangent Functions
Essential Questions: What is a function? How is the sine definition different from the sine function? Cosine? Tangent? From the graph of these functions, list some properties that describe them? Rebecca Adcock, a former student of EMAT 6690 at The University of Georgia, and I agree that the concept of the Sine, Cosine Functions will occur at lesson 6 of a beginning trigonometry unit. I praise her and her work because I want to use her thoughts on this particular lesson and build upon it with the tangent function.

Please notice what we mean by a function and connecting this with the values along the unit circle.
After Rebecca’s lesson, you should know exactly what the sine and cosine functions look like. Below is a summary of this information.
Sine Function

[pic]

Notice that the sine goes through the origin and travels to a maximum at (π/2, 1). Then, it travels down through (π, 0) to a minimum at (3π/2, -1). Finally the sine travels back up to (2π, 0). Then the sine wave will continue this same process again. Thus, the period of the sine function is 2π. Its amplitude is 1. Recall that sin (-x) = -sin x. This means that the sine function is odd, or it is symmetric to the origin.

Cosine Function

[pic]

Notice that the cosine goes through (0, 1), its maximum, to (π/2, 0) and down to (π, -1), its minimum. The cosine then travels back up through (3π/2, 0) and to (2π, 1). Then the cosine wave will continue this same process again. Thus, the period of the cosine function is also 2π. Its amplitude is 1. Recall that cos (-x) = cos x. This means that the cosine function is even, or it is symmetric to the y-axis.

Student Activity:
1. Give the domain and range of the sine and cosine functions. 2. What are the maximum and minimum values of these functions? 3. Identify the y-intercept and zeros of each function.
4. Identify which function is odd and which one is even....

...PROPERTIES OF SINE AND COSINEFUNCTIONS:
1. The sine and cosinefunctions are both periodic with period 2π.
2. The sinefunction is odd function since it’s graph is symmetric with respect to the origin, while the cosinefunction is an even function since it’s graph is symmetric with respect to y axis.
3. Thesinefunctions:
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 cosinefunction 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 cosinefunctions are continuous functions.
6. The domain of the sine and cosinefunctions is the set of all real numbers from -1 to 1
7. The amplitude of both the sine and cosinefunctions 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 cosinefunctions are 1 and -1 respectively, which occur alternately midway between the points where the functions is zero....

...architectures, presents a simulation of basic CORDIC cell and Implements Unfolded CORDIC Architecture on Spartan XC3S50 FPGA family. Keywords— CORDIC, Sine, Cosine, FPGA, CORDIC throughput
III. In Section IV we discuss the implementation of CORDIC algorithm in an FPGA and the simulation of basic CORDIC cell using Xilinx tool and XC3S50 Spartan3 family of FPGA is presented. The conclusion along with future research directions are discussed in Section V. II. CORDIC PRINCIPLE The CORDIC algorithm is based on the fact that any number may be represented by an appropriate alternating series. For example an appropriate value for e may be represented as e = 3- 0.3 + 0.02 - 0.002 + 0.0003 = 2.7183. The CORDIC technique uses a similar method of computation. There are two modes of operation for a CORDIC processor. a) Rotation Mode
I. INTRODUCTION Coordinate Rotation Digital Computer is abbreviated as CORDIC. Its implementation was first described in 1959 by Jack E. Volder [1], for the computation of trigonometric functions, multiplication and division. Further work has been carried out by J. S. Walther extending the use of CORDIC algorithm to calculate trigonometric, hyperbolic, logarithm, exponential and square root functions [2]. CORDIC large class of applications include: the generation of trigonometric, logarithmic and transcendental elementary functions, complex number multiplication, Eigen value...

...
CENTRAL INSTITUTE OF TECHNOLOGY, KOKRAJHAR
(Centrally Funded Institute under MHRD, Govt. of India)
KOKRAJHAR::783370:: BODOLAND
Estd. :: 2006
A
Project Report
On
SINE AND COSINEFUNCTION GENERATOR
USING VHDL
Submitted by,
DHARMESWAR BORO
ROLL NO: Gau-c-10/L-322.
PINOSH KR HAJOARY
ROLL NO: Gau-c-10/L-336.
MUNGSHAR BORO
ROLL NO: - Gau-c-10-267.
Table of contents:
CONTENTS Page No.
Title Page ………………………… 1
Candidate Declaration ………………………… 2
Certificate from the
Guidance ………………………… 3
Certificate from the Department ………………………… 4
Acknowledgement ………………………… 5
Abstract ………………………… 6
Chapter 1: Introduction 7
1.1 A general discussion applied algorithm. ………………………… 8-9
Chapter 2: Circuit Model ………………………… 10
2.1 Circuit Diagram ………………………… 10
2.2 Description of the circuit ………………………… 11
2.3 Divider Algorithm ………………………….. 12
2.4 Divider Flow Chart …………………………. 13
Chapter 3: Advantages and Disadvantages. …………………………. 14
3.1 Advantages …………………………. 14
3.2...

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Law of sines
In trigonometry, the law of sines (also known as the sine law, sine formula, or sine rule) is anequation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law,
where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right). Sometimes the law is stated using the reciprocal of this equation:
The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case.
The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines.
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Law of cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) is a statement about a general triangle that relates the lengths of its sides to the cosine of one of itsangles. Using notation as in Fig. 1, the law of...

...Section 5.2 Trigonometric Functions of Real Numbers
The Trigonometric Functions
EXAMPLE: Use the Table below to ﬁnd the six trigonometric functions of each given real number t. π π (a) t = (b) t = 3 2
1
EXAMPLE: Use the Table below to ﬁnd the six trigonometric functions 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 six trigonometric functions of each given real number t =
2
Values of the Trigonometric Functions
EXAMPLE: π π (a) cos > 0, because the terminal point of t = is in Quadrant I. 3 3 (b) tan 4 > 0, because the terminal point of t = 4 is in Quadrant III. (c) If cos t < 0 and...

...CIRCULAR FUNCTIONS
A different name of an angle is circular functions. Communicate the direction of a triangle to the length of the surface of a triangle. Trigonometric functions are important of triangles and form episodic occurrence, along with many complementary applications. Trigonometric functions have a wide range of uses including calculating indefinite lengths along with angles in triangles.
Trigonometricfunctions are normally specific as ratios of two sides of a right triangle including the angle, and able to equally specific as the lengths of different line segments from a unit circle.
More modern significance communicate them an infinite series or as solutions of specific different equations, allowing their extension to subjective positive also negative values and complex numbers. The sine with cosinefunctions are with usually used to model periodic function. Circular functions along angle θ are:
SineFunction:
sin θ = OppositeHypotenuse
CosineFunction:
cos θ = AdjacentHypotenuse
Tangent Function:
tan θ = OppositeAdjacent
Cosecant Function:
csc θ = HypotenuseOpposite = 1sinθ
Secant Function:
sec θ = HypotenuseAdjacent = 1cosθ
Cotangent Function:
cot θ = AdjacentOpposite = 1tanθ
A function of a...

...run,
all inputs are variable
3.1 The Production Function
Production function is a tool of analysis used in explaining the input-output relationship.
It describes the technical relationship between inputs and output in physical terms. In its
general form, it holds that production of a given commodity depends on certain specific
inputs. In its specific form, it presents the quantitative relationships between inputs and
outputs. A productionfunction may take the form of a schedule, a graph line or a curve,
an algebraic equation or a mathematical model. The production function represents the
technology of a firm.
An empirical production function is generally so complex to include a wide range of
inputs: land, labour, capital, raw materials, time, and technology. These variables form
the independent variables in a firm’s actual production function. A firm’s long-run
production function is of the form:
Q = f(Ld, L, K, M, T, t) (3.1.1)
where Ld = land and building; L = labour; K = capital; M = materials; T = technology;
and, t = time.
For sake of convenience, economists have reduced the number of variables used in a
production function to only two: capital (K) and labour (L). Therefore, in the analysis of
input-output relations, the production function is expressed as:
Q = f(K, L) (3.1.2)
Equation (3.1.2) represents the algebraic or...