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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 cosines states that

where γ denotes the angle contained between sides of lengths a and b and opposite the side of lengthc. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90° or π/2 radians), then cos(γ) = 0, and thus the law of cosines reduces to

The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. By changing which legs of the triangle play the roles of a, b, and c in the original formula, one discovers that the following two formulas also state the law of...

...APPLICATION OF SINELAW
The shorter diagonal of a parallelogram is 5.2 m. Find the perimeter of the parallelogram if the angles between the sides and the diagonal are 40o and 30o10’.
From the top of a 150 m lighthouse, the angles of depression of two boats on the shore are 20o and 50o, respectively. If they are due north of the observation point, find the distance between them.
SOLUTION
Solved for a
150/sin50 = a/sin90
a = (150 sin90)/sin50
a = 195.81 m
Solved for b
195.81/sin20 = b/sin30
b = (195.81 sin30)/sin20
b = 286.26 m
Therefore, the distance between the two ships is 286.26 m.
Two policemen 122 meters apart are looking at a woman on top of a tower. One cop is on the east side and the other on the west side. If the angles of elevation of the woman from the cops are 42.5o and 64.8o, how far is she from the two cops?
SOLUTION
Solved for a
122/sin72.7 = a/sin42.5
a = (122 sin42.5)/sin72.7
a = 86.33 m
Solved for b
122/sin72.7 = b/sin64.8
b = (122 sin64.8)/sin72.7
b = 115.62 m
Therefore, the distances of the woman from the two cops are 86.33 m and 115.62 m.
The angle between Rizal St. and Bonifacio St. is 27o and intersect at P.Jose and Andres leaves P at the same time. Jose jogs at 10 kph on Rizal St. If he is 3 km from Andres after 30 minutes,...

...not well known due to dominance of software systems over the past many years. Among these algorithms there is a simple shift-add algorithm known as CORDIC. CORDIC is being widely used in many domains like Image Processing, Communication, Robotics, Signal Processing applications due to its simple hardware efficient algorithm which is based on shift and add hardware. As CORDIC occupies less gate count in FPGA, it has been drawing attentions among many researchers and efforts have been made to improve its throughput and power keeping the constraints in mind. This paper summarizes the CORDIC 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...

...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 OF COS S
I...

...cos11π/12
cos(x/2) = ± √((1+cosx)/2) we use this Half-Angle Formula to evaluate it.
Since in this question, 11π/12 is in the Quadrant II, so cos(11π/12) should be negative.
cos11π/12 = – √((1+cos(11π/6))/2) = – √((1+(√3/2))/2)
to simplify it, both sides multiply by 2, – √(((1+(√3/2))(2))/((2)(2))) = – √((2+(√3))/4)
You should know what cos(11π/6) is, and you just plug in the number and you should get the answer.
(√3 – i)-10
We are using De Moivre’s Theorem to solve this problem.
De Moivre’s Theorem:
If z=r(cosθ + i sinθ), then for any integer n, zn=rn(cos(nθ) + i sin(nθ)).
So , we have z = √3 – i, and we would like to evaluate z-10 = (√3 – i)-10.
First, we need to express z = (√3 – i) into polar form.
r = √(〖(√3)〗^2+1^2 )=2
tanθ = -1/√3 θ = 5π/6
So, z=2(cos(5π/6) + i sin(5π/6))
Apply De Moivre’s Theorem, z-10 = (√3 – i)-10 =2-10 (cos(10*5π/6) + i sin(10*5π/6)) = ......
And, I think you should be able to get the answer for (√3 – i)-10.
19. Sketch triangle and find other five ratios of θ. sin θ = 3/5
Step 1, we sketch a right triangle to specify the angle θ.
Step 2, since we are given sin θ = 3/5, mark the opposite and hypotenuse as the graph below.
Step 3, find out the missing length by using Pathagorean Theorem. 32 + adjacent2 = 52 adjacent = 4.
Step 4, we can have all the other five trigonometric ratios:
sin θ = 3/5 Csc θ = 5/3
cos θ = 4/5 sec θ = 5/4
4sinƟcosƟ + 2√3 sinƟ – 2√3 cosƟ – 3 = 0...

...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.
...

...TASK 1
Explain the reference to legal principle and relevant case law, the legal aspect of placing the ‘Klick’ clock in the shop window with a price tag attached.
Ann antiques has a rare ‘Klick’ clock on its shop with price tags of €1,000 attached. In spite of its wording the sign in the window does not constitute a legal offer, it is merely an invitation to treat. Invitation to treat is an indication that the person who invite is willing to enter into a negotiation but it is not yet prepared to be bound. This case may be seen in Fisher v Bell (1961). It was held that having switch-blade knives in the window of a shop was not the same as offering them for sale.
TASK 2
Analyze the reference to legal principle and relevant because law, the legal effect of the event that transpired between Ann and Beth ignoring the conversation that took place between Carol and Beth and advice as to whether the valid contract exist between them.
The original invitation to treat at €1,000 was met by an offer from Beth which offers €500 on the ‘Klick’ clock. After Ann received an offer from Beth, Ann made a counter offer on the clock that she would sell €750 for it. It is up to Beth to decide whether to accept the offer or not. A counter offer arises when the offeree tries to change the terms of an original offer.
For example, the Hyde v Wrench (1940) case. In that case, on 6th June, Wrench offered to sell his estate to Hyde for £1,000 but...

...
CENTRAL INSTITUTE OF TECHNOLOGY, KOKRAJHAR
(Centrally Funded Institute under MHRD, Govt. of India)
KOKRAJHAR::783370:: BODOLAND
Estd. :: 2006
A
Project Report
On
SINE AND COSINE FUNCTION 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 Disadvantages...

...This judgment is subject to final editorial corrections approved by the
court and/or redaction pursuant to the publisher’s duty in compliance
with the law, for publication in LawNet and/or the Singapore Law
Reports.
BNJ (suing by her lawful father and litigation
representative, B)
v
SMRT Trains Ltd and another
[2013] SGHC 286
High Court — Suit No 432 of 2011
Vinodh Coomaraswamy JC (as he then was)
29–31 October 2012; 1–2, 5–9, 19–20 November 2012; 11 March 2013
Tort — Negligence — Breach of Duty
Tort — Occupier’s Liability — Who is an Occupier
Tort — Negligence — Res Ipsa Loquitur
Tort — Breach of Statutory Duty — Essential Factors
Contract — Contractual Terms — Implied Terms
31 December 2013
Judgment reserved
Vinodh Coomaraswamy J:
1
On 3 April 2011, a train coming into the Ang Mo Kio MRT station
(“AMK Station”) struck the plaintiff, causing her tragic and life-changing
injuries. She was then just fourteen years old. In these proceedings, she seeks
damages from two defendants for the injuries she suffered on that day. The
first defendant is SMRT Trains Ltd (“SMRT”). SMRT is a public transport
operator and holds the license to operate the mass rapid transit (“MRT”)
system along the North-South line. SMRT operates AMK Station and the train
which injured the plaintiff. The second defendant is the Land Transport
Authority of Singapore (“the LTA”). The LTA is a statutory board charged
BNJ v SMRT Trains Ltd...

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