solving a trigonometric equation sin(2x)=cos(x) with the double angle identity, precalculus tutorial

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Learn how to solve this trigonometric equation sin(2theta)=cos(theta) for your trig or precalculus class. We will use the double angle formula for sine and turn this into a quadratic equation.
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#blackpenredpen #math #trigonometry #apcalculus

blackpenredpen

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It shall be (2n+1)pi/6 where n=0 to 5, so we have 6 solutions within the range.
Sin(2t)=cos(t)=sin(pi/2-t)
=> 2t=n*pi+pi/2-t
=>t=(2n+1)pi/6
Since 0<=t<=2pi, 0<=n<=5

jigshere

if we divide cos(o) on both sides
2sin(o) = 1
sin(o) = 1/2
o = pi/3 + 2pik or 2pi/3 + 2pik

nemanjalazarevic

Thanks for the explanation. It helped a lot.

harhar

At 0:54 you could also just divide by cos(theta) on both sides to get 2sin(theta)=1, because cos(theta) can't be 0 because theta must be less than 2(pi), as you said it was in the thumbnail.

frede

How do you know to avoid dividing both sides by cos(theta) since it will hide two possible solutions?

JefiKnight

or you could have used the firmulas...
cosx = 0
cosx = cos(π/2)
so x = 2κπ + π/2
οr x = 2κπ - π/2, κ is integer
and you want
0 <= x <= 2π
so u get
π/2, 3π/2

sinx = 0.5
sinx = sin(π/6)
x = 2κπ + π/6
or x = 2κπ + π/6
// π - π/6 = 5π/6

and u get π/6
and 5π/6

theopapa

Hey, what's with the blue pen! Just joking, love your videos.

vvop

Thank you so much I am slowly getting the hang of it have a test on it tomorrow

beast_gago

in degrees :
sin(2x) = cos(x)
sin(2x) = sin(90 - x)
2x = 90 - x
3x = 90
x = 30

answer tau/12

AlgyCuber

Why you wrote under root 3 and under root -3? In base

tayyabasadiq

Nice video bro!
Can you explain me why is pi/6?

imanolmanzanares

[ 2sinQcosQ=cosQ => 2sinQ= 1 => sinQ=1/2 ] ?

virunkhoem

Solving simple question in a complex way.
See my solution:
sin(2©)=cos©
sin(2©)=sin(90°-©)
2©=90°-©
3©=90°
©=30°
Other values of© are(180°-30°)=150°
(360°-30°)=330°
(360°-150°)=210°

Shreyas_Jaiswal

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