What is Trigonometry identity: Definition and 13 Discussions

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. P

    B Question about orthogonal vectors and the cosine

    Hi, The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...] So...
  2. chwala

    Prove the trigonometry identity and hence solve given problem

    Refreshing on trig. today...a good day it is...ok find the text problem here; With maths i realize one has to keep on refreshing at all times... my target is to solve 5 questions from a collection of 10 textbooks i.e 50 questions on a day-day basis...motivation from late Erdos...
  3. A

    B Trigonometry Identity Question

    Can someone please tell me how sin(180 - x) = sin x? Here my attempt: sin (180 - x) = sin 180 . cos x - cos 180 . sin x Next? I have no idea...
  4. A

    Trigonometry Identity Question

    For the given triangle as below: I can obtain trigonometry identities as below: ##\sin α = \frac{y}{r} = \cos (90° - α)## ##\cos α = \frac{x}{r} = \sin (90° - α)## ##\tan α = \frac{y}{x} = \cot (90° - α)## ##\cot α = \frac{x}{y} = \tan (90° - α)## ##\sec α = \frac{r}{x} = \csc (90° - α)##...
  5. TyroneTheDino

    Integrating Using a Substituation

    Homework Statement Use the substitution ##u=\frac{\pi} {2}-x## evaluate the integral ##\int_0^\frac {\pi}{2} \frac {\sin x}{\cos x + \sin x}dx##. Homework Equations [/B] ##\cos (\frac {\pi}{2}-x)=\sin x## The Attempt at a Solution [/B]I start by plugging "u" into the equation making the...
  6. W

    Further Trigonometry Identity (Proving question)

    1) Question: Show that (sin3A-sinA)/(cosA+cos3A)=tanA 2) Relevant equations: tan A=sinA/cos A 1+tan^2A=sec^A cot A=1/tanA cot A=cos A/sinA sin^2A+cos^2A=1 secA=1/cos A cosecA=1/sinA 1+cosec^2A= cot^2A sin2A=2sinAcosA cos2A=1-2sin^2A=cos^2A-sin^2A=2cos^A-1 tan2A=(2tanA)/1-tan^2A 3)Attempt...
  7. J

    Heat equation problem so confusing

    Homework Statement The problem is f(x) = sin2πx - (1/πsquare)*sinπx and its given Bn sin (nπx) = f(x) Question is find Bn. Homework Equations Bn = 2/L ∫ (sin2πx - (1/πsquare)*sinπx) * sin(nπx/L) where L is 1 The Attempt at a Solution I did [/B] ∫ sin2πx * sin (nπx) - (1/πsquare)*sin...
  8. L

    Parametrics tangent line with point given

    Homework Statement I understand the setup for finding the slope, but always get confused whether I've fully simplified when trig identities get involved. [/B] Homework Equations My dy/dx is [/B] 4sin(θ)cos(θ) -2csc2(θ) which I simplified to just (-2sin(θ)cos(θ))/(csc2(θ) Does that...
  9. I

    Trigonometry identity sin(pi)cos(wpi)+cos(pi)sin(wpi)

    Homework Statement Hi guys, How can sin(∏)cos(ω∏)-cos(∏)sin(ω∏) = sin(ω∏)? please guide me trigonometry identity to apply with this? Homework Equations The Attempt at a Solution
  10. J

    How do I get from #2 to #3 in this trigonometry identity problem?

    I'm working on some trigonometry identity problems, and I'm stuck on this particular problem. This is how it is shown in the solution, but I cannot see how to get the highlighted part. Any help on how to get from #2 to #3 (the highlighted) is very much appreciated Jesper
  11. J

    Trigonometry Identity problem I been trying to solve all day

    Homework Statement solve 3cos(x) + 3 = 2 sin^2(x) where 0 <= x < 2piHomework Equations The Attempt at a Solution 3(cos(x) + 1) = 2 sin^2(x) 3(cos(x) + 1) = 2 (1- cos^2(x)) I've tried this variation, and a couple others but it just does not pan out. Please help. Oh yeah we have a real...
  12. F

    Help me prove this Trigonometry identity

    Someone pls help me prove this identity. I'm going nuts :confused: tan^2x - sin^2x = tan^2x sin^2x
  13. K

    Trigonometry identity proof trouble. (Help much appreciated)

    I can do pretty much all the proofs except these two, though I've tried to make a start. Note that in the questions the angle is given as theta, but for the sake of my keyboard I've changed it to angle 'x'. Q15: Prove: Tan2xsecx = 2sinxsec2x. So I need to show the two sides are equal...