Linear Combinations of Sinusoids

Slides:



Advertisements
Similar presentations
Trigonometric Functions
Advertisements

The arc length spanned, or cut off, by an angle is shown next:
Onward to Section 5.3. We’ll start with two diagrams: What is the relationship between the three angles? What is the relationship between the two chords?
The Unit Circle.
In these sections, we will study the following topics:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 366 Find the values of all six trigonometric functions.
10.3 Double-Angle and Half-Angle Formulas
Graphing (Method for sin\cos, cos example given) Graphing (Method for cot) Graphing (sec\csc, use previous cos graph from example above) Tangent Graphing,
Copyright © Cengage Learning. All rights reserved. Analytic Trigonometry.
1997 BC Exam. 1.6 Trig Functions Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008 Black Canyon of the Gunnison National.
CHAPTER 4 – LESSON 1 How do you graph sine and cosine by unwrapping the unit circle?
Ch 4 Trig Functions. 4.1 Radian and Degree Measures Converting from Radians to Degrees Converting from Degrees to Radians.
Using Trig Formulas In these sections, we will study the following topics: o Using the sum and difference formulas to evaluate trigonometric.
6.5 Double-Angle and Half-Angle Formulas. Theorem Double-Angle Formulas.
Warm up Use the Pythagorean identity to determine if the point (.623,.377) is on the circumference of the unit circle Using Pythagorean identity, solve.
WARM UP What is the exact value of cos 30°? What is the exact value of tan 60°? Write cos 57° in decimal form. Write sin 33° in decimal form. Write csc.
Trigonometry Section 7.3 Define the sine and cosine functions Note: The value of the sine and cosine functions depend upon the quadrant in which the terminal.
WARM UP For θ = 2812° find a coterminal angle between 0° and 360°. What is a periodic function? What are the six trigonometric functions? 292° A function.
Trigonometric Identities
Analytic Trigonometry
Sinusoids.
Graphs of Cosine Functions (part 2)
5.6 Phase Shift; Sinusoidal Curve Fitting
Introduction The Pythagorean Theorem is often used to express the relationship between known sides of a right triangle and the triangle’s hypotenuse.
DOUBLE-ANGLE AND HALF-ANGLE FORMULAS
9-3: Other Identities.
5 Trigonometric Identities.
WARM UP By composite argument properties cos (x – y) =
Addition and Subtraction Formulas
WARM UP What is amplitude of the graph? What is the period? 8
More Vector Basics.
Unit Circle and Radians
Copyright © Cengage Learning. All rights reserved.
5.3/5.4 – Sum and Difference Identities
Splash Screen.
Splash Screen.
Splash Screen.
VECTORS Dr. Shildneck.
VECTORS Dr. Shildneck.
Find sec 5π/4.
5.3 The Tangent Function.
Lesson 1: Establishing Trig Identities
Objectives Students will learn how to use special right triangles to find the radian and degrees.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
7.10 Single-Sided Exponential
Copyright © Cengage Learning. All rights reserved.
Double-Angle and Half-Angle Formulas 5.3
21. Sum and Difference Identities
Day 54 AGENDA: DG minutes No Unit Circle 4-function Calc only.
Copyright © 2017, 2013, 2009 Pearson Education, Inc.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Graphs of Sine and Cosine: Sinusoids
Trigonometry. Trigonometry More Trig Triangles.
Sum and Difference Identities
Section 4.5 Graphs of Sine and Cosine Functions
Copyright © Cengage Learning. All rights reserved.
15. Sum and Difference Identities
Other Trigonometric Identities
Sum and Differences Of Periodic Functions
Section 3 – Sum and Difference Identities
15. Sum and Difference Identities
7.3 Sum and Difference Identities
Graphs of Sine and Cosine Sinusoids
Section 5.5 Graphs of Sine and Cosine Functions
7.4 Periodic Graphs & Phase Shifts Objectives:
Vectors Lesson 4.3.
Objective: Use power-reducing and half angle identities.
5.4 Sum and Difference Identities
Sum and Difference Formulas (Section 5-4)
Presentation transcript:

Linear Combinations of Sinusoids Dr. Shildneck Fall 2014

Linear Combinations of Sinusoidal Functions Part I Linear Combinations of Sinusoidal Functions

The Sum of Two Sinusoids If you add two sinusoidal functions (wave functions) that are not in phase, the result will be another sinusoid with the same period and a phase shift. The amplitude of the result will be less than the sum of the amplitudes of the composed functions.

The linear combination of function, can be written as a single cosine function with a phase displacement (shift) in the form Where A = amplitude of the new wave, and D = the phase displacement.

Graphically, D is the shift of the sinusoidal curve , while A is the amplitude of the new curve.

On the unit circle, D is an angle, in standard position whose horizontal component is a and vertical component is b (which are the coefficients in the original combination). A (the amplitude) is the distance of the horizontal and vertical components of the combination. A b = 2 D a = 1 Q1. How can we use this information to find D? Use arctan(b/a). Q2. How can we use it to find A? Use the Pythagorean Theorem.

Example Write in terms of a single cosine function.

PROPERTY The Linear Combination of Sine and Cosine functions with equal periods, can be written as a single cosine function with phase displacement. Where and . Note: The signs of a and b specify the appropriate quadrant for D. A should be written in exact terms when possible. D can be rounded to 3 decimals.

Sum and Differences Of Periodic Functions Part II Sum and Differences Of Periodic Functions

Derive the Cosine of a Difference Using the Unit Circle to Derive the Cosine of a Difference

v θ = u - v u

θ

θ θ = u - v Since , we can write an equivalence relation for the lengths of the segments.

Derive the Cosine of a Sum Use the previous identity and even/odd identities to Derive the Cosine of a Sum

Derive the Sine of a Sum and The Sine of a Difference You can use the previous identities, co-function identities, and even/odd identities to Derive the Sine of a Sum and The Sine of a Difference

SUM and DIFFERENCE IDENTITIES

Example 1 Find the exact value of

Example 2 Find the exact value of

Example 3 Find the exact value of if , in Quadrant 1 and in Quadrant 2.

Example 4 Write as an expression of x.

Example 5 Solve on

ASSIGNMENT Alternate Text P. 395 #63-84(m3), 85, 87, 93-108(m3) Foerster P. 394 #1-9 (odd), 17, 23, 25