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POLAR COORDINATES. There are many curves for which cartesian or parametric equations are unsuitable. For polar equations, the position of a point is defined.
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10.1 Graph Sine, Cosine, and Tangent Functions
The Tangent Function.
Lectures in Microeconomics-Charles W. Upton The Keith-Leffler Model.
EXAMPLE 1 Solve a trigonometric equation Solve 2 sin x – 3 = 0. SOLUTION First isolate sin x on one side of the equation. Write original equation. 2 sin.
4.5 – Graphs of Sine and Cosine A function is periodic if f(x + np) = f(x) for every x in the domain of f, every integer n, and some positive number p.
Honors Pre-Cal Mrs. Cortez. xy = sin(x) 00 π /6½ π /21 5 π /6½ π 0 7 π /6-½-½ 3 π /2 11 π /6-½-½ 2π2π 0.
(1,0) (0,1) (-1,0) (0, -1) α (x,y) x y 1 sin(α) = y cos(α) = x (cos(α), sin(α)) (0,0) tan(α) = y/x 2.1 Unit Circle.
Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.
y x- intercepts x.
Y x- intercepts x. y x x- intercepts x y x y x y.
5.2-The Unit Circle & Trigonometry. 1 The Unit Circle 45 o 225 o 135 o 315 o 30 o 150 o 110 o 330 o π6π6 11π 6 5π65π6 7π67π6 7π47π4 π4π4 5π45π4 3π43π4.
State the amplitude and period for each function
Solving Trigonometric Equations Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x y π π 6 -7 π 6 π 6.
EXAMPLE 1 Use the formula for circumference Find the indicated measures. Write circumference formula. Substitute 9 for r. Simplify. Use a calculator. =
Students will graph the reciprocal trigonometric functions using transformations. Students will write equations of the reciprocal trigonometric functions.
7.7 Trig Functions of Real Numbers. Ex 1) Find the rectangular coordinates of P ( t ) a. P(0) b. c. (1, 0) P(t) = (x, y) on unit circle (0, –1) x y angle.
Reciprocal Trigonometric Functions. Reciprocal The Reciprocal of any number x is: Ex: Find the reciprocal of 0.5x 2 Equivalent forms of this definition:
Verify a trigonometric identity
EXAMPLE 1 Find trigonometric values Given that sin = and < < π, find the values of the other five trigonometric functions of . 4 5 π 2.
(7.4) Patterns, relationships, and algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form. The student.
AREA BOUNDED BY A POLAR CURVE. The area of the region bounded by a polar curve, r = f (θ ) and the lines θ = α and θ = β is given by: β α A = 1 2 r 2.
Verify a trigonometric identity
S/N 29. F(x)= sin (x-π/4) F(x)= sin x - 2 F(x)= sin (-x) F(x)= 2 sin x F(x)= sin x + 2 F(x)= - sin x F(x)= sin 2x F(x)= sin (x+π/4)
π Lesson 14.1, For use with pages
10.2 Translate and Reflect Trigonometric Graphs
Angle Identities. θsin θcos θtan θ 0010 –π/6–1/2√3/2–√3/3 –π/4–√2/2√2/2–1 –π/3–√3/21/2–√3 –π/2–10Undef –2π/3–√3/2–1/2√3 –3π/4–√2/2 1 –5π/6–1/2–√3/2√3/3.
Standardized Test Practice
13.2 – Define General Angles and Use Radian Measure.
Introduction to Unit Circle Trigonometry 1. The Unit Circle on the Coordinate Plane (1,0) (0,1) (-1,0) (0, -1) Quadrant I X – Pos Y - Pos X Y Radius =
30º 60º 1 45º 1 30º 60º 1 Do Now: Find the lengths of the legs of each triangle.
Sectors, Segments, & Annuli
EXAMPLE 4 Combine a translation and a reflection Graph y = –2 sin (x – ). 2 3 π 2 SOLUTION STEP 1Identify the amplitude, period, horizontal shift, and.
11-2 Surface Area of Prisms and Cylinders Objective: To find the surface area of a prism and a cylinder.
EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3.
CIRCULAR MEASURE. When two radii OA and OB are drawn in a circle, the circle is split into two sectors. The smaller sector OAB is called the minor sector.
Use the formula for circumference
Trigonometric Functions sine, cosine and tangent.
13-5 The cosine Function Today’s Objective: I can graph the cosine function.
Radians Introduction and Conversion of Angles. Mathematicians use a unit of measurement called radians to measure angles. Why you ask? Degrees & radians.
6.3 Graphing Trig Functions
Choose a category. You will be given the answer. You must give the correct question. Click to begin.
6.3 – GRAPHING SINE AND COSINE FUNCTIONS. Periodic Function and Period A function is periodic if, for some real number α, f(x + α) = f(x) for each x.
13-4 The Sine Function Today’s Objective: I can graph the sine function.
Interest rate (r ) Inflation rate (π) Fed Reaction Rule If the Fed sees the inflation rate rising… …it moves to raise interest rates r0r0 r1r1 π1π1 π0π0.
Integration of ex ∫ ex.dx = ex + k ∫ eax+b.dx = eax+b + k 1 a.
EXAMPLE 1 Graph a vertical translation Graph y = 2 sin 4x + 3. SOLUTION STEP 1 Identify the amplitude, period, horizontal shift, and vertical shift. Amplitude:
Medina Natural : Medina2 Any number that can be located somewhere on a number line Counting Number Counting numbers including the.
5.3 Solving Trigonometric Equations Day 2 Objective: In this lesson you will be learning how to solve trigonometric equations To solve a trigonometric.
Circles Formulas Circumference = π * diameter C = πd Area = π * r2
Pg. 323/361 Homework Pg. 362 #39 – 47 odd, 51, 52 Memorize Trig. Info #2QIV#4QIII#6QIII #8-250°#10470° #12338°, 698°, -382°, -742°#14210°, 570°, -510°,
Section 4.5 Graphs of Sine and Cosine. Sine Curve Key Points:0 Value: π 2π2π π 2π2π 1.
Chapter 11 Review Classwork Answers on last slide.
C HAPTER 14 D AY 6 Inverse Trigonometric Functions.
Phases of the Moon. Half of the moon is always illuminated.
Graphing Form of Sine and Cosine Functions
Solving Basic Trigonometric Equations Digital Lesson Section 5-3.
11.4 – Circumference and Arc Length. Circumference: C = dC = 2 r Length around a circle.
EXAMPLE 2 Graph a cosine function SOLUTION Graph y = cos 2 π x. 1 2 The amplitude is a = 1 2 and the period is 2 b π = 2 π = 2π2π 1. Intercepts: ( 1, 0)
© T Madas. r Area = π x radius A =A = π x rx r π = 3.14 [2 d.p.] special number it has its own name x radius x rx r A =A = π x r 2x r 2 How do we find.
EXAMPLE 1 Evaluate trigonometric expressions Find the exact value of (a) cos 165° and (b) tan. π 12 a. cos 165° 1 2 = cos (330°) = – 1 + cos 330° 2 = –
Ch 6.7 – Graphing Other Trig Functions. y = cscx Period: Domain: Range: Asymptotes: y = 1: y = -1: 2π2π All real numbers except πn, n is an integer All.
Circles Shape and Space. The value of π For any circle the circumference is always just over three times bigger than the radius. The exact number is called.
Trigonometry Using Co-terminal and Reference Angles Dr. Shildneck Fall, 2014.
Holt McDougal Geometry 12-3-EXT Measuring Angles in Radians 12-3-EXT Measuring Angles in Radians Holt Geometry Lesson Presentation Lesson Presentation.
Exact Values of Sines, Cosines, and Tangents None.
Radian Measure. What is to be learned What a radian is How to convert between radians and degrees.
PreCalculus 5-3 Solving Trigonometric Equation. Trigonometric Equations To solve trigonometric equations, we must solve for all values of the variable.
Surface area The surface area of a solid is found by adding the areas of all its faces. To calculate the surface area you may first sketch the net of the.
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