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Trigonometric identitiesTrigonometryHigh schoolabout 1:19

Is sin(x + pi) = sin(x) an identity?: animated lesson

A narrated, step-by-step trigonometric identities video lesson with captions. Final answer: This is not an identity: the two sides are not equal for all values..

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    sin⁡(x+π)=sin⁡(x)\sin\left( x+\pi\right) = \sin\left( x\right)
  2. 2. Steps, part 1
    1. sin⁡(θ+π)=−sin⁡θ, cos⁡(θ+π)=−cos⁡θ\sin(\theta + \pi) = -\sin\theta,\ \cos(\theta + \pi) = -\cos\theta
      Rewrite everything in terms of sin x and cos x
    2. −sin⁡x-\sin x
      Left side in sine and cosine
    3. sin⁡x\sin x
      Right side in sine and cosine
    4. LHS−RHS=ND,N=−2sin⁡x\text{LHS} - \text{RHS} = \frac{N}{D},\quad N = -2\sin x
      Subtract the sides and write the difference over a common denominator
  3. 3. Steps, part 2
    1. N=−2sin⁡xN = -2\sin x
      Replace every cos²x with 1 − sin²x and collect terms
    2. The numerator is not zero, so the sides are not identical
  4. 4. The answer
    sin⁡(x+π)≢sin⁡(x)\sin\left( x+\pi\right) \not\equiv \sin\left( x\right)

    LHS − RHS numerator = -2·sin(x)

  5. 5. Check
    • At x = 30°, the left side ≈ -0.5 but the right side ≈ 0.5.

Transcript

The problem. Here is the problem: sin(x plus pi) equals sin(x). Prove a trigonometric identity.

Steps, part 1. Rewrite everything in terms of sin x and cos x. Identities used:. Left side in sine and cosine. Right side in sine and cosine. Subtract the sides and write the difference over a common denominator. The identity is true exactly when this numerator is zero for every value of the variable.

Steps, part 2. Replace every cos squaredx with 1 minus sin squaredx and collect terms. The numerator is not zero, so the sides are not identical. At x equals 30°, the left side approximately negative 0.5 but the right side approximately 0.5.

The answer. So the answer is This is not an identity: the two sides are not equal for all values. The difference of the sides does not reduce to 0. At x equals 30°, the left side approximately negative 0.5 but the right side approximately 0.5. The equation may still be true for particular values; solve it as an equation to find them.

Check. At x equals 30°, the left side approximately negative 0.5 but the right side approximately 0.5.

About trigonometric identities

Decide whether a trigonometric equation is an identity and show why.

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Is sin(x + pi) = sin(x) an identity?: Step-by-Step Lesson