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i) sin² x = -(1−cos 2x) (I used this to no- ii) cos²x == (1 + cos2x) - 2 1 iii) sin a sinb= 2 functions are orthogonal) 1 cos(a - b)-cos(a + b) 2

i) sin² x = -(1−cos 2x) (I used this to no- ii) cos²x == (1 + cos2x) - 2 1 iii) sin a sinb= 2 functions are orthogonal) 1 cos(a - b)-cos(a + b) 2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.4: Multiple-angle Formulas
Problem 69E
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From the previous take home you showed that e 2-ix = cos x - i sinx. The equation retix = r(cos x ±isin x) first appeared in Euler's Introductio in 1748. From these equations it is easy to show that: eix -e-ix sin x = and that cos x = e +e 2i 2 Using these relationships and the rules that apply to multiplying exponentials, show that the following relationships are true: 1 ii) cos²x = - (1 + cos2x) 2 -ix i) sin²x = (1-cos 2x) (I used this to normalize the PIAB wave functions in class) 2 2 1 iii) sin a sin b = - 2 functions are orthogonal) cos(a−b) cos(a+b) (this relationship is useful to show that PIAB wave NOTE: The approach you used to solve these three problems can be used to get other useful trigonometric relations of this type and these relations can come in handy for simplifying many problems.
Transcribed Image Text:From the previous take home you showed that e 2-ix = cos x - i sinx. The equation retix = r(cos x ±isin x) first appeared in Euler's Introductio in 1748. From these equations it is easy to show that: eix -e-ix sin x = and that cos x = e +e 2i 2 Using these relationships and the rules that apply to multiplying exponentials, show that the following relationships are true: 1 ii) cos²x = - (1 + cos2x) 2 -ix i) sin²x = (1-cos 2x) (I used this to normalize the PIAB wave functions in class) 2 2 1 iii) sin a sin b = - 2 functions are orthogonal) cos(a−b) cos(a+b) (this relationship is useful to show that PIAB wave NOTE: The approach you used to solve these three problems can be used to get other useful trigonometric relations of this type and these relations can come in handy for simplifying many problems.
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