Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

haleyking345

  • one year ago

Solving Trig Equations a.) sec x + tan x=1

  • This Question is Closed
  1. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I don't know what to do.

  2. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Recall\[\sec x=\frac{1}{\cos x}\\\tan x = \frac{\sin x}{\cos x}\] \[\sec x+\tan x=1\\ \frac{1}{\cos x}+\frac{\sin x}{\cos x}=1\\ \frac{1+\sin x}{\cos x}=1\\ 1+\sin x=\cos x\] Use the identity \[\sin^2x+\cos^2x=1\] \[1+\sqrt{1-\cos^2x}=\cos x\\ \sqrt{1-\cos^2x}=\cos x-1\\ 1-\cos^2x=(\cos x-1)^2\\ 1-\cos^2x=\cos^2x-2\cos x+1\] Does that help?

  3. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Would it be cot=1?

  4. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well, that last equation simplifies to \[2\cos^2x-2\cos x=0\] What can you do with that?

  5. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do you add 2 cos x to the other side?

  6. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Then divide by 2

  7. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sure, you could do that, but then you get \[\cos^2x=\cos x\] Then your first instinct would be to divide both sides by a cosx, which would eliminate a potential solution. Here's a hint: What's a common factor of 2cos²x and 2cosx?

  8. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    2? Or is it cos?

  9. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Actually, it's both!

  10. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Factor out a 2cosx from both terms, and you get \[2\cos x \;(\cos x - 1)=0\] What's next?

  11. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Add 1 to the other side?

  12. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    No, that's not it. Think of it this way: When you have \[a\cdot b = 0, \text{ either $a$ or $b$ (or both) must be zero, right?}\] This means that for 2cosx (cosx - 1) to be equal to 0, either \[2\cos x=0 \text{ or } \cos x-1=0\]

  13. haleyking345
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh so whhen they are in paranthesis, you can set them equal to 0?

  14. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Parentheses aren't exactly the qualifying factor here, no. When you have the product of two things equal to 0, such as \[(ax+b)(cx+d)=0, \text{ then you can split up the left side and get}\\ ax+b=0 \text{ AND }cx+d=0.\\ \text{Solving both equations gives you $x=-\frac{b}{a}$ and $x=-\frac{c}{d}$ as solutions.}\]

  15. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The parentheses are there to show that the term (cosx - 1) is a factor of 0, as is 2cosx.

  16. SithsAndGiggles
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    There's a big difference between \[2\cos x\;(\cos x-1) \text{ and } 2\cos x \cos x -1\]

  17. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.