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. I

    Trigonometric Equation: Solve for x on Real Numbers Set | Homework Statement

    Homework Statement cos(x)+2tg(x) = 7/(4*cosx(x)) Solve for x on the real numbers set. Homework Equations tg(x) = sin(x)/cos(x) , cos^2(x) = 1-2*sin^2(x) , The determinant is D = b^2 - 4ac , Also cos(x) ,tg(x), sin(x) | -1 < x < 1 The Attempt at a Solution For final arrangement I...
  2. P

    Solving a Trigonometric Equation to Hit a Moving Target

    So the orginial question asked what angle to fire a gun to hit a moving target, and i got all the way to 40sinθcosθ-10sinθ-1 = 0, but we aren't supposed to use calculators so how would you solve this by hand. Thanks!
  3. C

    Solve x: Trig Equation (4/3)sinXcosX-sin22X=0

    Homework Statement Solve for x. x is greater than or equal to 0 and x is less than or equal to 2pi. (4/3)sinxcosx-sin22x=0 If there is a better way to approach this question than the way I did it below, please post it. Thanks.Homework Equations N/A The Attempt at a Solution...
  4. K

    How can I solve these trigonometric equations?

    Homework Statement The original problem is this: ab^2*cos(2x)*(cos(x))^2 + ba^2*sin(2x)*(sin(x))^2 = 0 And some of the more "logical" forms I've got to are these:2[tan(x)]^3-2tan(x) = -b/a tan(2x)*(tan(x))^2=-b/a But I can't solve these equations so I'd like to ask for some help here.
  5. James889

    Solving Trigonometric Equation with Cosine and Sine Identities

    Hi, I would like som help on this one: Give all the real soulutions to the equation \frac{7}{4}-2sin(x) - cos^2(x) = 0 i tried the common identities, like replacing cos^2(x) with 1-sin^2(x) any ideas?
  6. T

    Another Trigonometric Equation

    Homework Statement Solve for t sin3tcost+cos3tsint=-1/2 Homework Equations Sum and Difference Identities The Attempt at a Solution I saw this problem the the sum identity sin(\alpha+\beta) looked like the way to begin solving for t. This condenses to sin(3t+t)= -1/2 which...
  7. A

    Solving simultaneous trigonometric equation

    how to solve eqn1 = sin((alpha)*a)*cosh((alpha)*b)=sin(alpha)*a eqn2 = sinh((alpha)*b)*cos((alpha)*a) = b*sin(alpha) alpha = 0 to 2pi... Find a & b... Plz help me with any method that would help solving this...
  8. X

    Solving a Trigonometric Equation: (cos x)^4 = cos (x^2)

    Hi all, I was tutoring someone in Algebra III today and encountered a problem that threw me for a loop. (cos x)^4 = cos (x^2) Is there a way to solve this algebraically?
  9. L

    Simplify trigonometric equation problem

    Homework Statement Simplify the following. Write all trigonometric functions in terms of t. Homework EquationsThe Attempt at a Solution I know that: sec(t+2pi)=sec(t) 1+tan(t+3\pi)= 1+tan(t) What about: csc(t-6\pi)? Will it be equal to csc(t). Can anyone please help me with this? Thank you...
  10. A

    How Do You Solve Trigonometric Equations Involving Cosine and Sine?

    Hi All, I’m currently attempting to work ahead on my precalculs and I’m looking at trigonometric equations. I seem to have a bit of a problem with this example (i haven’t had too many issues with the rest of the exercises), 9cos(2x)+sin(x)=9 (solve for x in the interval 0 <=x <=2pi.) I’m...
  11. I

    Trigonometric Equation with Multiple Functions and Arguments

    Can anyone help solve the below for 'a' 0=b(ycos(a)-bcos(2a)+xsin(a)) I've reduced it to b=ycos(a)+2sin(a)sin(a)+xsin(a) =sin(a)(cot(a)+2sin(a)+x) Which I think is right, but I can't get any further without just ending up with different forms that appear more complex.
  12. N

    Solving trigonometric equation

    Hi guys, I need to solve the following trigonometric equation. Some time ago I would have used Fortran with the appropriate libraries but now I was wondering whether there is any better approach. Any suggestion? w^2/c^2= k//^2 - Sqrt(w^2 Eps_b^2/c^2 - k//^2) * Tan[Sqrt(w^2 Eps_b^2/c^2 -...
  13. M

    Solving a trigonometric equation help

    Homework Statement sqrt3 cot 3x + 1 = 0 where 0 _<x_<2pie Homework Equations The Attempt at a Solution I narrowed it down to a constant on one side, cot 3x = -1/sqrt3 which I could turn into cos x/sin x = -1/sqrt3 but I have no idea where to go from here. Any help...
  14. A

    Solving a trigonometric equation

    I'm trying to solve: sin2x = -cos2x so my first step was... 2sinxcosx = -cos^2x - sin^2x and then... 2sinxcosx + cos^2x + sin2x = 0 but I'm confused on how I would end up factoring 2sinxcosx + cos^2x + sin2x, or if I did something wrong earlier in the problem...thank you
  15. S

    Trigonometric Equation problem

    Homework Statement If cos x = a/b and tan x = c/d; where a, b, c do not equal 0, then sin x is... (a) bd/ac (b) bc/ad (c) ad/bc (d) ad/bc Homework Equations The Attempt at a Solution The way I see it, if cos x = a/b, then b=hypotenuse. if tan x = c/d, then d=...
  16. I

    Exact value of Trigonometric equation.

    Homework Statement 1/\sqrt{}2 cos 15 - 1/\sqrt{}2 sin 15 The Attempt at a Solution I got \sqrt{}3/2. If anyone could confirm this, I would appreciate it. Thanks.
  17. H

    Help solving trigonometric equation?

    Can someone please give me some tips on how to solve this? : 400*(2,2*cosV-1)*tanV=20*9,81, I am supposed to find V, but I can't see how to solve this equation.
  18. P

    Draw the triangle with sides 1, 3, and sqrt(10)

    Evaluate sin(tan^{-1}(\frac{1}{3})) 3. The Attempt at a Solution Let y = tan^{-1}(\frac{1}{3}) <=> tan(y)=1/3 y = ? sin(tan^{-1}(\frac{1}{3})) = sin y I think for solution to exist tan^{-1}(\frac{1}{3}) must be in the range of sin, (that is \left\frac{-\pi}{2}...
  19. T

    Can You Solve This Trigonometric Equation 2 Homework Problem?

    Homework Statement tan^2x = 2tanx*sinx Homework Equations N/A The Attempt at a Solution tan^2x = 2tanx*sinx (tan^2x)/tanx = (2tanx*sinx)/tanx tanx = 2sinx sinx/cosx = 2sinx cosx = 1/2 x = pi/3 \alpha = pi/3 and 5pi/3 Not sure what I'm doing wrong. The...
  20. I

    Fairly basic trigonometric equation

    Acos(A*x)-Bcos(B*x)=0, where A>B Is there a general solution for an equation of this form? Thanks, intangible
  21. S

    Solving a trigonometric equation

    Homework Statement Determine the points of intersection for the two given functions on the interval 0<x<4pi y = 2 \sin \frac{x}{2} y = 3 \cos \frac{x}{3} 2. The attempt at a solution Well i tried graphing it and found out that the solution must lie somewhere between pi and 2pi...
  22. S

    Proving trigonometric equation?

    for x between 0 and 2pie, solve cos(4x)=sin(2x)... is this a proving trigonometric equation? i don't think it is
  23. J

    PLease HELP with this trigonometric equation

    Greetings my friends: I have been reading a book about optimization and I found the following trigonometric equation: tan(10x.pi)= - 10 pi x (this equation goes from x E [-1,2] it is easy to see that has infinite solutions, but the author came to the conclusion that the solutions are...
  24. C

    Complicated trigonometric equation

    I need to solve sin(ax)sin(bx) - k cos(ax)cos(bx) = -1 where k > 1 is constant and so are a and b, and they are all irrational. (It's part of a much larger question in optics...) Any ideas, please? Thanks
  25. S

    Factoring trigonometric equation

    Hi, I'm having trouble understanding how my math book factored the following equation. [(cos(t)sin(t) - (1 + sin(t))cos(t)) / (cos(t)cos(t) - (1 + sin(t))sin(t))] to get [(cos(t)(1+2sin(t))) / ((1 + sin(t)) (1 - 2sin(t)))] I get the numerator but I don't understand what happened...
  26. P

    Can this trigonometric equation be solved for x_1 and x_2 in terms of \alpha?

    x_1(cos\alpha-1) + x_2sin\alpha = 0 x_1sin\alpha + x_2(-cos\alpha-1) = 0 How to solve this equation? Can anyone help me?
  27. A

    Derive a trigonometric equation for the volume of the cone

    A circular cone is inscribed in a sphere with a radius of 30cm. The semi vertical angle is theta. Derive a trigonometric equation for the volume of the cone. This has be stumped. I tried looking up proofs for the expression of the volume of a cone for inspiration but all involve calculus.
  28. C

    Solving a Trigonometric Equation: Primary to Secondary Solutions

    How do I get from the primary to the secondary solution of a trigonometric equation? This book tells me that the second angle is within -\pi \leq \theta \leq \pi, in a different quadrent, but I don't follow :\ Thanks. Edit: I got it (I think!): I can pick the correct quadrent using the...
  29. N

    Solving a Trigonometric Equation

    How would you solve for \alpha in the following equation? 4=\cos(\alpha)+\cos^2(\alpha)+\cos^4(\alpha)
  30. C

    How do I solve a trigonometric equation on a given interval?

    Hello all Just refreshing in maths, and want to know if I am doing this correctly: Solve the equation \sin x = 0.2 on the interval 0 \leq x < 2\pi . I took \arcsin(0.2) , however how do you solve for all solutions in the interval? (i.e what if the interval had been 0 \leq x < \pi...
  31. J

    How to Solve a Trigonometric Equation with Multiple Solutions?

    How to solve this equation: tan 2x + sec 2x = cos x + sin x where 0<=x<=360 I solve it in this way but cannot find the right answer: (sin 2x + 1) / cos 2x = cos x + sin x (cos x + sin x)^2 / cos 2x = cos x + sin x (cos x + sin x) / cos 2x = 1 sqrt(2) cos(x - 45) sec 2x = 1 sec 2x cos(x -...
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