Trigonometric equation Definition and 131 Threads

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

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  1. Eclair_de_XII

    Can someone check my solution to a trigonometric equation?

    Homework Statement 4sec2θ tanθ = 8 tanθ Homework Equations sec2θ = tan2θ + 1 The Attempt at a Solution 4(tan2 + 1) tanθ = 8 tanθ (4tan2 + 4) tanθ = 8 tanθ 4 tan3 + 4 tanθ = 8 tanθ 4 tan3 - 4 tanθ = 0 4tanθ(tan2θ - 1) = 0 4tanθ = 0 tanθ = 0 tan-1(tanθ) = tan-1(0) tan-1θ = 0, π tan2θ - 1 = 0...
  2. chwala

    Solving a trigonometric equation

    Find the value of x between 0 degrees and 360 degree satisfying the equation 10sin^2x+ 10sin x cos x - cos^2x = 2 this is how i have attempted..... 10 sin^x+ 10sin2x/2 - cos^x = 2 I used the property sin 2x = 2 sin x cos x and substituted sin x cos x with sin 2x/2 giving me.... 11sin^2x +...
  3. anemone

    MHB Solve the trigonometric equation

    Solve the equation $\sin^6 a+\cos^6 a=0.25$.
  4. A

    MHB Solve the trigonometric equation?

    a) Solve the trigonometric equation: A = cosX + AsinX, for some angle X. b) The imaginary number i is equal to √-1. Use part a) to solve i = cosX + isinX. You will have used degrees to answer parts a) and b). However, the mathematics below requires the use of radians. Convert degrees to...
  5. A

    Trig Equation: Is This Procedure Correct?

    Is correct the following procediment? ## A \sin (\omega t) = A \sin (\phi) \to \phi= \sin^{-1} \sin (\omega t )## Is correct to say that ## \phi = \omega t## is oscillatory in this case ??
  6. S

    How do I find the remaining solutions for the trigonometric equation?

    Homework Statement 4sin2(2x) -1 = 0 Solve over 0 --> 2pi The attempt at a solution (2sin(2x) + 1) (2sin(2x) - 1) = 0 2sin(2x) +1 = 0 sin(2x) = -1/2 2x = -pi/6 x = -pi/12 and -11pi/12 which = 23pi/12 and 13pi/12. 2sin(2x) - 1 = 0 sin(2x) = 1/2 2x = pi/6 x= pi/12 and...
  7. MarkFL

    MHB Trentan's question at Yahoo Answers regarding a trigonometric equation

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  8. anemone

    MHB Show the trigonometric equation has no solutions

    Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.
  9. anemone

    MHB Solve a trigonometric equation

    Let $y$ be in radians and $0<y<\dfrac{\pi}{4}$. Solve for $y $ if $\tan 4y=\dfrac{\cos y-\sin y}{\cos y +\sin y}$.
  10. karush

    MHB Trigonometric equation: 2cos(θ) + 2sin(θ) = √(6)

    $2\cos{\theta}+2\sin{\theta}=\sqrt{6}$ $\displaystyle\cos{\theta}+\sin{\theta}=\frac{ \sqrt{6} }{2}$ $\displaystyle(\cos{\theta}+\sin{\theta})^2=\frac{3}{2}$ $\displaystyle\cos^2{\theta}+2cos\theta\sin\theta+\sin{\theta}^2=\frac{3}{2}$ $\displaystyle\sin{2\theta}=\frac{1}{2} \Rightarrow...
  11. G

    Trigonometric Equation: 2sin(2x) = 2cos(x)

    Homework Statement 2 sin(2x) = 2 cos(x) Homework Equations The Attempt at a Solution 4sinxcosx-2cosx=0 2(2sinxcosx-cosx)=0 2cosx(sinx-1)=0 2cosx=0 sinx=1 And then I did the math and got pi/2 and 3pi/2 , but that's apparently wrong.
  12. B

    Trigonometric equation with tangent

    Homework Statement Solve: 2 tan x (tan x - 1) = 3. Homework Equations Pythagorean identities? The Attempt at a Solution I tried the following: 2 tan^2 x - 2 tan x = 3 2 (sec^2 x - 1) - 2 tan x = 3 2 (1 - cos^2 x) - 2 sin x cos x = 3 cos^2 x (multiplying through by cos^2 x)...
  13. anemone

    MHB How Can You Solve Complex Trigonometric Equations?

    Solve the equation \sqrt{3}\sin x(\cos x-\sin x)+(2-\sqrt{6})\cos x+2\sin x+\sqrt{3}-2\sqrt{2}=0
  14. MarkFL

    MHB Emma's question at Yahoo Answers regarding solving a trigonometric equation

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  15. anemone

    MHB Solve A Trigonometric Equation

    Solve \sqrt{2}\cos \left(\frac{x}{5}-\frac{\pi}{12} \right)-\sqrt{6}\sin \left(\frac{x}{5}-\frac{\pi}{12} \right)=2\left(\sin \left(\frac{x}{5}-\frac{2\pi}{3} \right)-\sin \left(\frac{3x}{5}+\frac{\pi}{6} \right) \right)
  16. anemone

    MHB Solve a trigonometric equation

    Solve the equation \cos^2 \left(\frac{\pi}{4}(\sin x+\sqrt{2}\cos^2 x)\right)-\tan^2 \left(x+\frac{\pi}{4}\tan^2 x\right)=1
  17. B

    Trigonometric Equation solving

    Homework Statement cos 2x + cos 4x = cos x 0 degrees ≤ X ≤ 360 degrees Homework Equations Trig identities. The Attempt at a Solution Trig identities and equations are my weakness so I'm trying to review some to get better at them, so I guess this isn't technically homework...
  18. MarkFL

    MHB Answerer's question at Yahoo Answers regarding a trigonometric equation

    Here is the question: Here is a link to the question: PRE-CALC QUESTION!? HELP !? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  19. MarkFL

    MHB Eaglesfan1717's question at Yahoo Answers regarding a trigonometric equation

    Here is the question: Here is a link to the question: Help with trig equation :)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  20. Fernando Revilla

    MHB Katie's question at Yahoo Answers: trigonometric equation

    Here is Katie's question: Here is a link to the original question: Solve for x in the equation 1+sinx/cosx + cosx/1+sinx = 4 for 0 < x < 2pi? - Yahoo! Answers I'll post a link there to this topic so the OP can find my response.
  21. L

    MHB Proving a trigonometric equation

    I want to prove this equation: \cos\left(\frac{2\pi}{5}\right)+\cos\left( \frac{4\pi}{5}\right)+\cos\left( \frac{6\pi}{5}\right)+\cos\left(\frac{8\pi}{5} \right) = -1 . I've no idea how to begin. I think it may be related to complex numerd but I don't know how to combine them.
  22. U

    Inverse trigonometric equation

    Homework Statement Solve cot^{-1} x - cot^{-1} (n^2 - x+1)=cot^-1(n-1) Homework Equations The Attempt at a Solution I can write the above equation as θ+α=β where the symbols represent the respective inverse functions. Now I take tan of both sides. Simplifying I get...
  23. K

    Solving a trigonometric equation

    Homework Statement Solve 2cos(x)^2 + 3cos(x) + 1 = 0 for 0 <= x <= 2pi Homework Equations Trigonometric equations, yadda yadda yadda. The Attempt at a Solution 2cos(x)^2 + 3cos(x) + 1 = 0 cos(x)(2cos(x) + 3) = -1 cos(x) = -1/(2cos(x) + 3) I then figured out that you get a solution when...
  24. H

    MHB Solving a Trigonometric Equation

    2Cos (3/10x) - 3Sin(1/2x)=0 How would I go about solving this equation? I do not even know where to begin.
  25. M

    MHB Solving a Trigonometric Equation for Cos (x)

    Hi, may you help me? I have a question I solved but couldn't ended. If cos(x)+5Sec(x)=1.1 then Cos(x)=? 2Cos(x)+5Sec(x) =11/10 2Cos(x)+5/Cos(x)=11/10 2Cos(x)*Cos(x)+5=11/10*Cos(x) 2Cos^2(x)+5=11/10*Cos(x) 2Cos^2(x)-11/10-Cos(x)-5=0 then i used delta=b^2-4ac then i found delta=-4.79...
  26. T

    MHB Solving a Trigonometric Equation

    solve for tan^4x + tan^2x = sec ^4x - sec^2x i solved and ended up with RIHS= tan^4x?
  27. alane1994

    MHB Solving a Trigonometric Equation

    I have this as part of a calculus problem. I guess I am a little rusty on this. Any help would be appreciated. 4\cos^2{x}=\frac{\sec^2{x}}{4} I believe that I need to solve for cosx. And then determine what cosx is in radians. That should be the lower limit of the integration. I know how it...
  28. O

    Solving a general complex trigonometric equation

    Hello everyone! Homework Statement Solving a relatively general equation representing a constant combination of 2 trigonometric functions.Homework Equations a\cos{(vx+p)}+b\cos{(ux+q)}=cThe Attempt at a Solution I really don't have any idea for a general solution to this equation.. Best Regards
  29. Feodalherren

    Trigonometric Equation: Solving Sin 2x = -2cos x in the Interval 0 ≤ x < 2∏

    Homework Statement All solutions in the interval 0 ≤ x < 2∏ Sin 2x = -2cos x The Attempt at a Solution I went about it in two different ways. The first way: I simply used the double angle formula to get 2sinx cosx + 2cos x = 0 From this we get that whatever makes cos x = 0...
  30. Z

    Dumbfounded by this trigonometric equation.

    Homework Statement A beach ball is riding the the waves near Tofino, BC. The ball goes up and down with the waves according to the formula h= 1.4 sin(∏t/3), where h is the height in meters above sea level, and t is the time in seconds. A) In the first 10s when is the ball @ sea level. B)...
  31. A

    Finding x in an inverse trigonometric equation

    Homework Statement Given y = 2/3(arctan \frac{x}{4}), find x. Homework Equations Inverse trigonometric identities for tan. (I guess)The Attempt at a Solution I never attempted because it seems impossible for me. You think this is possible? Thank you!
  32. A

    Inverse Trigonometric Equation problem

    EDIT: 1.Homework Statement The problem asks to simplify this: cos(arcsin x)Homework Equations let arc sin x = θThe Attempt at a Solution EDIT: I have to edit because I received a warning for not posting for #3 and #2. I originally posted "cos(arcsin x)" for #2 because that was what I know...
  33. J

    MHB Number of Real Solutions for Trigonometric Equation

    Number of real solution of the equation $\sin (\sin (\sin x)) = \cos (\cos (\cos x))$ where $ 0 \leq x\leq 2\pi$
  34. S

    MHB Need Help With a Trigonometric Equation?

    can somebody help me out on this one,i got stuck on this one https://www.physicsforums.com/attachments/164
  35. M

    Trigonometric Equation: Solving tan(\alpha)=\sqrt{2}-1 for \alpha in ]0,90°[

    Homework Statement tan(\alpha)=\sqrt{2}-1 for every alpha in ]0,90°[ Homework Equations 1-count tan(2α) 2-conclude the value of α The Attempt at a Solution 1-after using tan(2α)=(2tanα)/1-tan^2α) i got tan(2α)=(√2-1)/2 2- I know that α is pi/8 but i just don't know how to...
  36. lep11

    Mastering Trigonometric Equations: Solving cos 2x-cos^2 x=0

    Homework Statement cos 2x-cos^2 x=0 The Attempt at a Solution I have no idea.
  37. F

    Solving a complex trigonometric equation

    Hi, thank you for looking. Homework Statement Solve the following complex equation and carefully plot at least 3 solutions for each. State ALL solutions to each! State exact answers! Sin(z)=√(3) The Attempt at a Solution Here is my work. I was a little confused how to tell what...
  38. M

    Can a Trigonometric Equation Be Solved Algebraically?

    Hi, I wolud like to solve the folowing equation: A*cos(B*t)+sin(B*t)*(C-t)-D=0 (t is unknown) It is urgent. Thanks in advance. Regards Mike
  39. E

    Finding the tangent and normal of a trigonometric equation?

    Homework Statement a. find the tangent and normal line at P (0,0) b. find when the tangent is horizontal Equation: 3x+sin3x at (0,0) The Attempt at a Solution Find the derivative- y'=3 (?) or it it 3x+cos(3x)? But from there, I don't know how to find the rate of change to create a...
  40. Saitama

    Solving a Trigonometric Equation: Sec Theta

    Homework Statement This is a question from my test paper. If sec \theta=\frac{4xy}{(x+y)^2}, where x,y \in R, then (a)x,y \in R+ (b)x,y \in R- (c)x=y\neq0 (d)x\inR+, y\inR- Homework Equations The Attempt at a Solution I don't understand what i have to do and where should...
  41. nukeman

    Having Trouble: Solving a Trigonometric Equation using Identities ?

    Having Trouble: Solving a Trigonometric Equation using Identities ? Homework Statement I am having problems with these. Need some help please! Here is a sample question I am having troubles with: 3 cos θ + 3 = 2 sin^2 θ, 0 <= θ < 2pi I am really struggling with these. Any help...
  42. I

    Can following trigonometric equation be simplified:

    Fx = sin [ 90 - theta ] * [ dX - [tan (theta) * dY ] ] ??
  43. B

    Trigonometric Equation Homework Help: Solving 3sinx + 4cosx = 1

    Homework Statement I took a test in my Precalculus class about 3 weeks ago, and the tests were handed back this week. I saw that I didn't do so well because I was stumped on this equation: 3sinx + 4cosx = 1 Homework Equations We were to use the following formulas to solve this equation...
  44. T

    Problem with finding multiple solutions to a trigonometric equation

    Homework Statement Find all solutions to the following equation: sin((5*x/2)+15)=0.433 where 0 < x < 360 Homework Equations The Attempt at a Solution i can find the first value of x that satisfies this equation. (5*x/2)+15=sin^-1(0.433) 5*x/2=10.6581 5*x=21.2162 x=4.2632...
  45. C

    Solving for theta in a Trigonometric Equation

    Homework Statement Tan \Theta = 12 / 5, sin\Theta<0Homework Equations Sin ^{2} \Theta+ Cos ^{2}\Theta = 1 The Attempt at a Solution Sin is less than 0 so... it should be somwhere in the II or III quadrant? What identities should I use to solve?
  46. ThomasMagnus

    Trigonometric Equation: Solving for Theta with Quadratic Formula

    Solve for \theta 20= 32[sin\theta] (72/32[cos\theta]) + 1/2(-9.8){(72/32[cos\theta]}2I'm really having trouble with this question. I've tried it many times, but keep getting stuck. Is there anyway to let X=72/32[cos\theta] and then solve for x with the quadratic equation? Can someone help me...
  47. P

    Trigonometric Equation: Solving for Sec x=2 and Reviewing for a Test

    This is a problem from a quiz I had awhile ago, I got the answer correct, but I'm reviewing for a test and I completely forget how I did it. Can someone please show me what I did? sec4x=2 secx=2 x=π/3 + 2nπ ---> x=π/12+2nπ/4 x=5π/3 + 2nπ ---> x=5π/12+2nπ/4 π=pi p.s. I mostly just forget how...
  48. S

    Problem with a trigonometric equation.

    Homework Statement Which of the following is not a possible solution of 0=sinθ+cosθtan^2θ? A. 3pi/4 B. 7pi/4 C. 2pi D. 5pi/2 Homework Equations All trigonometric identities. The Attempt at a Solution Too much to write down, at least two pages long, but I've been trying to...
  49. X

    Trigonometric Equation problem

    I'm stuck on a question from my calculus homework...T_T Here's the question... There is a large clock at the front of the lounge on the paddle steamer. The hour hand is 28 cm long and takes 12 hours to rotate once. At 2 o'clock the tip of the hour hand is 195 cm above the floor. There's...
  50. T

    Solving trigonometric equation

    This is part of the identity proving question . from \sin C=\frac{2}{\sqrt{29}} , how can i reach C=\frac{1}{2}\sin^{-1} (\frac{20}{29}) ?
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