Trigonmetric equations help

kelleynicole30

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Jul 17, 2008
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Find the solutions of the equation that are in the interval [0,2?). Please show steps (especially rearranging the equation to solve it. Thanks for helping me. I know the solutions are 11?/6, ?/2 for the first problem and 7?/6, 11?/6 for the second problem.


1A) 1-sint=(?3)cost
B) 2cos?+tan?=sec?
 
kelleynicole30 said:
Please show steps (especially rearranging the equation to solve it.
We already know how to do these, so we don't really need the practice. And you already have worked examples in your textbook and in your class notes, so another worked example or two probably isn't going to make much difference in your understanding. (I'm assuming you're wanting to learn how to do this on your own, rather than hoping somebody will just "do" your homework for you.)

So please reply with your thoughts and efforts so far, even if you're fairly certain that you're on the wrong track. You might actually be on the right track, and just needing a little nudge. But we can't know that until we can "see" where you are.

Please be complete. Thank you! :D

Eliz.
 
kelleynicole30 said:
Find the solutions of the equation that are in the interval [0,2?). Please show steps (especially rearranging the equation to solve it. Thanks for helping me. I know the solutions are 11?/6, ?/2 for the first problem and 7?/6, 11?/6 for the second problem.


1A) 1-sint=(?3)cost

3cos(t)+1sin(t)=1\displaystyle \sqrt{3}\cdot cos(t) + 1\cdot sin(t)\, = \, 1

\(\displaystyle \sqrt{\sqrt{3}^2 + 1^2}\cdot[{\frac{\sqrt{3}}{\sqrt{\sqrt{3}^2 + 1^2}}\cdot cos(t) + \frac{1}{\sqrt{\sqrt{3}^2 + 1^2}}sin(t)]\, = \, 1\)

\(\displaystyle 2\cdot[{\frac{\sqrt{3}}{2}\cdot cos(t) + \frac{1}{2}sin(t)]\, = \, 1\)

[cos(π6+2nπ)cos(t)+sin(π6+2nπ)sin(t)]=12\displaystyle [cos(\frac{\pi}{6}+2\cdot n \cdot \pi)\cdot cos(t) + sin(\frac{\pi}{6}+2\cdot n \cdot \pi)\cdot sin(t)]\, = \, \frac{1}{2}

continue from here...


B) 2cos?+tan?=sec?

use

tan(α)=sin(α)cos(α)\displaystyle tan(\alpha) \, = \, \frac{sin(\alpha)}{cos(\alpha)}

and

sec(α)=1cos(α)\displaystyle sec(\alpha) \, = \, \frac{1}{cos(\alpha)}

continue from here...
 
Thank you so much for helping me understand this a whole lot better. :D I just couldn't figure out what to do but I realized that I needed to square both sides first.
 
Hello, kelleynicole30!

Solve in the interval [0,2π)\displaystyle \text{Solve in the interval }\,[0,\,2\pi)

. . B)    2cosα+tanα  =  secα\displaystyle B)\;\;2\cos\alpha + \tan\alpha \;=\;\sec\alpha

We have:   2cosα+sinαcosα  =  1cosα\displaystyle \text{We have: }\;2\cos\alpha + \frac{\sin\alpha}{\cos\alpha} \;=\;\frac{1}{\cos\alpha}

Multiply by cosα ⁣:    2cos2 ⁣α+sinα  =  1\displaystyle \text{Multiply by }\cos\alpha\!:\;\;2\cos^2\!\alpha + \sin\alpha \;=\;1

Then we have:   2(1sin2 ⁣α)+sinα  =  1\displaystyle \text{Then we have: }\;2(1-\sin^2\!\alpha) + \sin\alpha \;=\;1

. . which simplifies to:   2sin2 ⁣αsinα1  =  0\displaystyle \text{which simplifies to: }\;2\sin^2\!\alpha - \sin\alpha - 1 \;=\;0

. . which factors:   (sinα1)(2sinα+1)  =  0\displaystyle \text{which factors: }\;(\sin\alpha-1)(2\sin\alpha + 1) \;=\;0

\(\displaystyle \text{and has roots: }\;\begin{array}{ccccccc}\sin\alpha - 1 \:=\:0 & \Rightarrow &\sin\alpha \:=\:1 & \Rightarrow & \alpha \:=\:\frac{\pi}{2} & \text{extraneous root} \\ \\[-3mm] 2\sin\alpha + 1 \:=\:0 & \Rightarrow &\sin\alpha \:=\:-\frac{1}{2} & \Rightarrow & \boxed{\alpha \:=\:\frac{7\pi}{2},\:\frac{11\pi}{2}} \end{array}\)

 
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