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Properties of Trig Functions

Maths: Edexcel A Level Pure Maths, topic 5.2.2

Properties of the Trigonometric Functions

We can also use the unit circle to investigate the symmetry and repetition of the trigonometric functions.

Symmetry

Symmetry

  • Consider the coordinates of a point created by angle θ\theta and the coordinates of the point created by −θ-\theta.
  • The xx-coodinate is unchanged, so cos⁡−θ=cos⁡θ\cos-\theta = \cos\theta.
  • The yy-coordinate changes sign, so sin⁡−θ=−sin⁡θ\sin-\theta = - \sin\theta.
Symmetry

Symmetry

  • To determine the symmetry of the tangent function, we substitute −θ-\theta for θ\theta:
  • tan⁡−θ=sin⁡−θcos⁡−θ=−sin⁡θcos⁡θ=−tan⁡θ\tan-\theta = \frac{\sin-\theta}{\cos-\theta} = \frac{-\sin\theta}{\cos\theta} = -\tan\theta
Beyond 2π

Beyond 2π

  • For angles of θ\theta greater than 2π2\pi, we go back around to the positive xx-axis.
  • This means each point on the circle can be generated by many different values of θ\theta, namely:
  • θ+2nπ\theta +2n\pi where n=0,1,2,3,...n = 0,1,2,3,....
  • This means we can say that the sine, cosine and tangent are periodic functions on the range 0<θ<2π0<\theta<2\pi.
Beyond 2&pi;

Beyond 2π

  • The tangent function is actually periodic over the range 0<θ<π0<\theta<\pi.
  • This is because the negative sign of the coordinated in the third quadrant cancel each other to give:
  • −y−x=tan⁡(θ+π)=tan⁡θ\frac{-y}{-x} = \tan(\theta+\pi) = \tan\theta

Test yourself on Properties of Trig Functions

  • How often does the function tan⁡θ\tan\theta repeat itself?

  • How often does the function sin⁡θ\sin\theta repeat itself?

  • How often does the function cos⁡θ\cos\theta repeat itself?

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