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à 3.5ïSums of Trigonometric Functions
äPlease find the periods of the following sums of trigonome-
êëtric functions.
â
êêêë Find the period of
êêêè y = 3∙sin 2x + cos 2x
êêêè (Please see Details)
éSïMany physical situations can be described by trigonometric
functions.ïFor example, the current in an alternating circuit can be
described by a sine or cosine function.ïThe charge on a capacitor can
be described by a sine or cosine function.ïThe motion of a mass attach-
ed to a spring, sound waves, heartbeat rhythm, the gravitational force
of the moon acting on tides, periodic forcing functions, and many other
physical situations can be described by trigonometric functions.
è There are times, however, when you need two or more trigonometric
functions to describe the effects of waves from more than one source.
For example, suppose you have current being supplied from two power
sources.ïEach source could be described by a different function, and
the net result would be the sum of the two individual functions.ïIf you
strike two tuning forks with nearly the same frequency, you will hear
a combination of the two soundwaves that sound like a series of loud and
soft beats.ïIf you look at the effects of both the sun and the moon on
tides, you have two contributors causing a net result of tides described
by the sum of the two individual trigonometric functions.ïThus, there
are many physical situations that can be described by a sum of two or
more trigonometric functions.
è You are encouraged to experiment on your Function Plotter with func-
tions of the form y = A∙cos B(x + C) + D∙sin E(x + F).ïDraw A∙cosB(x+C)
and D∙sinE(x+F) individually, then draw the sum so that you can com-
pare the results.ïStart with simple curves like y = cos(x + π/2) + sinx
and y = cos(x - π/2) + sinx.ïTry to predict the results.ïSometimes
waves cancel each other out, and sometimes they work together to increase
the amplitude.
è The period of y = 3∙sin 2x + cos 2x is 2π/2 or π.
1êêèFind the period of
êêêïy = 3∙sin 2x + 4∙cos 3x
è A)ï2πêèB)ïπêè C)ï6πêèD)ïå of ç
ü
ë The period of 3∙sin 2x is 2π/2 or π.ïThe period of 4∙cos 3x is
2π/3.ïThe least common multiple of ç two periods is 2π.ïThis is
the period of the sum of the two individual functions.
Ç A
2êêèFind the period of
êêêïy = 3∙sin 7x + 3∙cos 8x
è A)ï56π/8êB)ï2πêèC)ï56πêïD)ïå of ç
ü
ë The period of 3∙sin 7x is 2π/7, and the period of 3∙cos 8x is
2π/8.ïThe least common multiple of ç two periods is 2π.ïThis is
the period of the sum of the two individual functions.
Ç B
3êêèFind the period of
êêêïy = 2∙sin 4x/3 + cos 2x
è A)ï8π/3ê B)ï2πêèC)ï3πêèD)ïå of ç
ü
ë The period of 2∙sin 4x/3 is 3π/2, and the period of cos 2x is
π.ïThe least common multiple of ç two periods is 3π.ïThis is
the period of the sum of the two individual functions.
Ç C
4êêèFind the period of
êêêïy = 3∙sin x/2 + 2cos 2x/3
è A)ï2πêèB)ï12πêïC)ï3πêèD)ïå of ç
ü
ë The period of 3∙sin x/2 is 4π, and the period of 2∙cos 2x/3 is
3π.ïThe least common multiple of ç two periods is 12π.ïThis is
the period of the sum of the two individual functions.
Ç B