Trigonometric Functions

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Hy po e te t n C i u s se o b a p p O θ A c B Adjacent Math 1060 ~ Trigonometry 4 The Six Trigonometric Functions Learning Objectives In this section you will: • Determine the values of the six trigonometric functions from the coordinates of a point on the Unit Circle. • Learn and apply the reciprocal and quotient identities. • Learn and apply the Generalized Reference Angle Theorem. • Find angles that satisfy trigonometric function equations. The Trigonometric Functions In addition to the sine and cosine functions, there are four more. Trigonometric Functions: y Ex 1: Assume � is in this picture. P(cos(�), sin(�)) Find the six trigonometric functions of �. x 1 1 Ex 2: Determine the tangent values for the first quadrant and each of the quadrant angles on this Unit Circle. Reciprocal and Quotient Identities Ex 3: Find the indicated value, if it exists. a) sec 30º b) csc c) cot (2) d) tan �, where � is any angle coterminal with 270 º. e) cos �, where csc � = -2 and < � < . f) sin �, where tan � = and � is in Q III. 2 Generalized Reference Angle Theorem The values of the trigonometric functions of an angle, if they exist, are the same, up to a sign, as the corresponding trigonometric functions of the reference angle. More specifically, if α is the reference angle for θ, then cos θ = ± cos α, sin θ = ± sin α. The sign, + or –, is determined by the quadrant in which the terminal side of θ lies. Ex 4: Determine the reference angle for each of these. Then state the cosine and sine and tangent of each. a) -225º b) c) - Finding Angles that Satisfy Cosine and Sine Equations Ex 5: Find all of the angles on the unit circle which satisfy the given equation. a) sin � = 0 b) cos � = - c) sin � = Finding Angles that Satisfy Other Trigonometric Equations Ex 6: Find all of the angles on the unit circle which satisfy the given equation. a) tan � = -1 b) sec � = 2 c) cot � = 0 3.

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Unit Circle Trigonometry
UNIT CIRCLE TRIGONOMETRY The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is xy22+ =1. A diagram of the unit circle is shown below: y xy22+ = 1 1 x -2 -1 1 2 -1 -2 We have previously applied trigonometry to triangles that were drawn with no reference to any coordinate system. Because the radius of the unit circle is 1, we will see that it provides a convenient framework within which we can apply trigonometry to the coordinate plane. Drawing Angles in Standard Position We will first learn how angles are drawn within the coordinate plane. An angle is said to be in standard position if the vertex of the angle is at (0, 0) and the initial side of the angle lies along the positive x-axis. If the angle measure is positive, then the angle has been created by a counterclockwise rotation from the initial to the terminal side. If the angle measure is negative, then the angle has been created by a clockwise rotation from the initial to the terminal side. θ in standard position, where θ is positive: θ in standard position, where θ is negative: y y Terminal side θ Initial side x x Initial side θ Terminal side Unit Circle Trigonometry Drawing Angles in Standard Position Examples The following angles are drawn in standard position: y y 1. θ = 40D 2. θ =160D θ θ x x y 3. θ =−320D Notice that the terminal sides in examples 1 and 3 are in the same position, but they do not represent the same angle (because x the amount and direction of the rotation θ in each is different).
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4.2 – Trigonometric Functions: the Unit Circle
Warm Up Warm Up 1 The hypotenuse of a 45◦ − 45◦ − 90◦ triangle is 1 unit in length. What is the measure of each of the other two sides of the triangle? 2 The hypotenuse of a 30◦ − 60◦ − 90◦ triangle is 1 unit in length. What is the measure of each of the other two sides of the triangle? Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr. Niedert 1 / 27 4.2 { Trigonometric Functions: The Unit Circle Pre-Calculus Mr. Niedert Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr. Niedert 2 / 27 2 Trigonometric Functions 3 Domain and Period of Sine and Cosine 4 Evaluating Trigonometric Functions with a Calculator 4.2 { Trigonometric Functions: The Unit Circle 1 The Unit Circle Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr. Niedert 3 / 27 3 Domain and Period of Sine and Cosine 4 Evaluating Trigonometric Functions with a Calculator 4.2 { Trigonometric Functions: The Unit Circle 1 The Unit Circle 2 Trigonometric Functions Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr. Niedert 3 / 27 4 Evaluating Trigonometric Functions with a Calculator 4.2 { Trigonometric Functions: The Unit Circle 1 The Unit Circle 2 Trigonometric Functions 3 Domain and Period of Sine and Cosine Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr. Niedert 3 / 27 4.2 { Trigonometric Functions: The Unit Circle 1 The Unit Circle 2 Trigonometric Functions 3 Domain and Period of Sine and Cosine 4 Evaluating Trigonometric Functions with a Calculator Pre-Calculus 4.2 { Trig Func: The Unit Circle Mr.
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The Unit Circle 4.2 TRIGONOMETRIC FUNCTIONS
292 Chapter 4 Trigonometry 4.2 TRIGONOMETRIC FUNCTIONS : T HE UNIT CIRCLE What you should learn The Unit Circle • Identify a unit circle and describe its relationship to real numbers. The two historical perspectives of trigonometry incorporate different methods for • Evaluate trigonometric functions introducing the trigonometric functions. Our first introduction to these functions is using the unit circle. based on the unit circle. • Use the domain and period to Consider the unit circle given by evaluate sine and cosine functions. x2 ϩ y 2 ϭ 1 Unit circle • Use a calculator to evaluate trigonometric functions. as shown in Figure 4.20. Why you should learn it y Trigonometric functions are used to (0, 1) model the movement of an oscillating weight. For instance, in Exercise 60 on page 298, the displacement from equilibrium of an oscillating weight x suspended by a spring is modeled as (− 1, 0) (1, 0) a function of time. (0,− 1) FIGURE 4.20 Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure 4.21. y y t > 0 (x , y ) t t < 0 t θ (1, 0) Richard Megna/Fundamental Photographs x x (1, 0) θ t (x , y ) t FIGURE 4.21 As the real number line is wrapped around the unit circle, each real number t corresponds to a point ͑x, y͒ on the circle. For example, the real number 0 corresponds to the point ͑1, 0 ͒. Moreover, because the unit circle has a circumference of 2␲, the real number 2␲ also corresponds to the point ͑1, 0 ͒.
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4.2 Trigonometric Functions: the Unit Circle
4.2 Trigonometric Functions: The Unit Circle The two historical perspectives of trigonometry incorporate different The Unit circle: xy221 2 2 3 1 1 3 Example: Verify the points , , , , , ,(1,0) are on the unit circle. 2 2 2 2 2 2 Imaging that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in the following. As the real number line is wrapped around the unit circle, each real number t corresponds to a point (,)xy on the circle. For example, the real number corresponding to (0,1) . 2 Remark: In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t . With this interpretation of t , the arc length formula sr (with r 1 ) indicates that the real number t is the (directional) length of the arc intercepted by the angle , given in radians. In the following graph, the unit circle has been divided into eight equal arcs, corresponding to t -values 3 5 3 7 of 0, , , , , , , ,2 4 2 4 4 2 4 Similarly, in the following graph, the unit circle has been divided into 12 equal arcs, corresponding to t 2 5 7 4 3 5 11 values of 0,,,, , ,, , , , , ,2 6 3 2 3 6 6 3 2 3 6 The Trigonometric Functions. From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t . You can use these coordinates to define the six trigonometric functions of t .
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4.1 Unit Circle Cosine & Sine (Slides 4-To-1).Pdf
The Unit Circle Many important elementary functions involve computations on the unit circle. These \circular functions" are called by a diﬀerent name, \trigonometric functions." Elementary Functions But the best way to view them is as functions on the circle. Part 4, Trigonometry Lecture 4.1a, The Unit Circle Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 1 / 54 Smith (SHSU) Elementary Functions 2013 2 / 54 The Unit Circle The Unit Circle The unit circle is the circle centered at the origin (0; 0) with radius 1. The radius of the circle is one, so P (x; y) is a vertex of a right triangle Draw a ray from the center of the circle out to a point P (x; y) on the with sides x and y and hypotenuse 1. circle to create a central angle θ (drawn in blue, below.) By the Pythagorean theorem, P (x; y) solves the equation x2 + y2 = 1 (1) Smith (SHSU) Elementary Functions 2013 3 / 54 Smith (SHSU) Elementary Functions 2013 4 / 54 Central Angles and Arcs Central Angles and Arcs An arc of the circle corresponds to a central angle created by drawing line segments from the endpoints of the arc to the center. The Babylonians (4000 years ago!) divided the circle into 360 pieces, called degrees. This choice is a very human one; it does not have a natural mathematical reason. (It is not \intrinsic" to the circle.) The most natural way to measure arcs on a circle is by the intrinsic unit of measurement which comes with the circle, that is, the length of the radius.
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Development of Trigonometry Medieval Times Serge G
Development of trigonometry Medieval times Serge G. Kruk/Laszl´ o´ Liptak´ Oakland University Development of trigonometry Medieval times – p.1/27 Indian Trigonometry • Work based on Hipparchus, not Ptolemy • Tables of “sines” (half chords of twice the angle) • Still depends on radius 3◦ • Smallest angle in table is 3 4 . Why? R sine( α ) α Development of trigonometry Medieval times – p.2/27 Indian Trigonometry 3◦ • Increment in table is h = 3 4 • The “ﬁrst” sine is 3◦ 3◦ s = sine 3 = 3438 sin 3 = 225 1 4 · 4 1◦ • Other sines s2 = sine(72 ), . Sine differences D = s s • 1 2 − 1 • Second-order approximation techniques sine(xi + θ) = θ θ2 sine(x ) + (D +D ) (D D ) i 2h i i+1 − 2h2 i − i+1 Development of trigonometry Medieval times – p.3/27 Etymology of “Sine” How words get invented! • Sanskrit jya-ardha (Half-chord) • Abbreviated as jya or jiva • Translated phonetically into arabic jiba • Written (without vowels) as jb • Mis-interpreted later as jaib (bosom or breast) • Translated into latin as sinus (think sinuous) • Into English as sine Development of trigonometry Medieval times – p.4/27 Arabic Trigonometry • Based on Ptolemy • Used both crd and sine, eventually only sine • The “sine of the complement” (clearly cosine) • No negative numbers; only for arcs up to 90◦ • For larger arcs, the “versine”: (According to Katz) versine α = R + R sine(α 90◦) − Development of trigonometry Medieval times – p.5/27 Other trig “functions” Al-B¯ırun¯ ¯ı: Exhaustive Treatise on Shadows • The shadow of a gnomon (cotangent) • The hypothenuse of the shadow
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Math 175 Trigonometry Worksheet We Begin with the Unit Circle. The
Math 175 Trigonometry Worksheet We begin with the unit circle. The definition of a unit circle is: x 2 + y 2 =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of a circle measured in the counterclockwise direction that subtends an arc length equal to 1 radius. Notice that the angle does not change with the radius. There are approximately 6 radius lengths around the circle. That is, one complete turn around the circle is 2π ≈ 6.28 radians. Define the Sine and Cosine functions: Choose P(x,y) a point on the unit circle where the terminal side of θ intersects with the circle. Then cosθ = x and sinθ = y . We see that the Pythagorean Identity follows directly from these definitions: x 2 + y 2 =1 (cosθ)2 + (sinθ)2 =1 we know it as : sin2 θ + cos2 θ =1 Example 1. Example 2. Determine: Determine: sin(90°) and cos(90°) sin(3π) and cos(3π) π Recall that90° corresponds to radians. (How many degrees do 3π radians correspond to?) 2 We can read the answers from the graphs: ⎛ π ⎞ sin(90°) = sin⎜ ⎟ = y coordinate of P =1 sin(3π) = sin(540°) = y coordinate of P = 0 ⎝ 2 ⎠ ⎛ π ⎞ cos(90°) = cos⎜ ⎟ = x coordinate of P = 0 cos(3π) = cos(540°) = x coordinate of P = −1 ⎝ 2 ⎠ Problems 1 and 2: 1. Locate the following angles on a unit circle and find their sine and cosine. 5π 5π a. − b. c. 360° d. −π 2 2 2.
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Trigonometry Review with the Unit Circle: All the Trig. You'll Ever Need
Trigonometry Review with the Unit Circle: All the trig. you’ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs: domain, range and transformations. Angle Measure Angles can be measured in 2 ways, in degrees or in radians. The following picture shows the relationship between the two measurements for the most frequently used angles. Notice, degrees will always have the degree symbol above their measure, as in “452° ”, whereas radians are real number without any dimensions, so the number “5” without any symbol represents an angle of 5 radians. An angle is made up of an initial side (positioned on the positive x-axis) and a terminal side (where the angle lands). It is useful to note the quadrant where the terminal side falls. Rotation direction Positive angles start on the positive x-axis and rotate counterclockwise. Negative angles start on the positive x-axis, also, and rotate clockwise. Conversion between radians and degrees when radians are given in terms of “ π ” π DEGREES Æ RADIANS: The official formula is θθD ⋅=radians 180D π 2π Ex. Convert 120D into radians Æ SOLUTION: 120D ⋅= radians 180D 3 180D RADIANS Æ DEGREES: The conversion formula is θ radians⋅ = θ D π π π 180DD 180 Ex. Convert into degrees. Æ SOLUTION: ⋅==36D 5 55π For your own reference, 1 radian≈ 57.30D A radian is defined by the radius of a circle. If you measure off the radius of the circle, then take the straight radius and curved it along the edge of the circle, the angle this arc marks off measures 1 radian.
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The Unit Circle
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Trigonometric Functions: the Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry is based on the Greek words for triangle measurement. Today, trigonometry has much broader applications than that of the measurement of angles and distances, now including the study of sound, light, and electrical waves. This section introduces trigonometry in terms of what is called the wrapping function, which takes the real number line and wraps it around the unit circle. The unit circle is a circle having a radius value of 1 and its center at the origin of a rectangular coordinate system. For example, if u and v are the variables in a rectangular coordinate system, the equation of the unit circle would be given by u2 + v2 = 1, and the graph of the circle would be as shown below. The origin or zero value of the real number line in the wrapping function is at the point (1,0) on the unit circle. Positive values of the line move around the circumference of the circle in a counterclockwise direction and negative values in a clockwise direction, as shown below. We would like to associate any point on the real number line with a corresponding point on the unit circle, called a circular point. We know that the circumference of a circle is equal to the diameter of the circle times π, or 2rπ, where r is the radius of the circle. For the unit circle, r = 1, so the circumference of the circle is equal to 2π.
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Complexity Reduction in the CORDIC Algorithm by Using Muxes
Yuhang Sun Yuhang Complexity Reduction in the CORDIC Algorithm by using MUXes Master’s Thesis Complexity Reduction in the CORDIC Algorithm by using MUXes Yuhang Sun Series of Master’s theses Department of Electrical and Information Technology LU/LTH-EIT 2015-460 Department of Electrical and Information Technology, http://www.eit.lth.se Faculty of Engineering, LTH, Lund University, August 2015. Complexity Reduction in the CORDIC Algorithm by using MUXes By Yuhang Sun Department of Electrical and Information Technology Faculty of Engineering, LTH, Lund University SE-221 00 Lund, Sweden 1 2 Abstract Nowadays, the CORDIC algorithm plays an important role to deal with the non-linear functions in hardware. In this thesis, a novel methodology is described to reduce the complexity in an unrolled CORDIC architecture, which gives higher speed, lesser area, and lower power consumption. That is, MUXes are used to replace adder stages. Five different unrolled CORDIC architectures have been implemented in ASIC using a 65nm CMOS technology with Low Power High ்ܸ transistors. The area, computational speed, accuracy, error behavior, and power consumption have been analyzed. The design aim is to reduce the power consumption, which is more and more important depending on the area. As a result the area and power consumption get 7.9% lower and 27.2% lower separately, and the speed is 22.9% higher compared to the original unrolled CORDIC architecture. Keywords: CORDIC, power consumption. 3 Acknowledgments This Master’s thesis would not exist without the support and guidance of many people, my friends, my parents, especially my supervisors. I would like to express my deepest gratitude to my supervisors, Professor Peter Nilsson, Erik Hertz, and Rakesh Gangarajaiah who guided me and helped me in this thesis work.
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Recreate the Unit Circle
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