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Help!! C2 trigonometry

ok, please help me to answer this question!!

Solve for theta, in ther interval 0<theta<360 means less than or equal to), the following equations:

4sintheta = tantheta

ok so i changed the tantheta identity to sintheta/costheta.

then multiplied the costheta up, to give:

4sinthetacostheta = sintheta

then divided by sintheta both sides to give:

4costheta = 1

and did inverse cos of (1/4) and got an answer of 75.5.

and because this is positive, and cos is positive in the 4th quadrant you do 360 - 75.5 to get 283.5.

both 75.5 and 283.5 are correct answers, but the other answers are 0, 180 and 360?! i would really appreciate it if someone could explain how they got this?!!!!

tks.
Reply 1
Avatar for Dekota-XS
Dekota-XS
2
4sinxcosx = sinx
=> 4sinxcosx - sinx = 0
=> sinx(4cosx-1) = 0

sinx = 0, x = arcsin0,
4cosx-1 = 0, x = arccos(1/4)
Reply 2
Avatar for Gemini92
Gemini92
Don't divide through by sinθsin\theta as you lose the solutions when sinθ=0sin\theta = 0
Reply 3
Avatar for dysons3
dysons3
OP
oh right, ok thankyou soo much!! also, one other question i didnt understand was, tan^2 (theta - 45) = 1?

i did inverse tan of (root 1) = 45, and then i know you add 45 to this to get 90 which is one of the answers :smile: but then other answers include 0, 180, 270, 360?

tks!
Reply 4
Avatar for Dekota-XS
Dekota-XS
2
Tan(x) is periodic with a period of 90, so tanx = t, tan(x+90) = t, tan(x-90) = t, tan(x+90+90) = t etc

If 90 is a solution, then so is 90+90, 90-90, 90+90+90 and finally 90 x 4.

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