01.5-Complementary

Complementary Angles

Recall that two angles are complementary if they add to $90^\circ \big( \text{ or } \dfrac{\pi}{2} \big)$

For example, $60^\circ \text{ and } 30^\circ$ are complementary because $60^\circ + 30^\circ = 90^\circ$

In radians: $\dfrac{\pi}{3} + \dfrac{\pi}{6} = \dfrac{\pi}{2}$

Complementary Angles in Trigonometry

The sine of an angle is equal to the cosine of its complement.

  • $\sin(\theta) = \cos(90^\circ - \theta)$
  • $\sin(\theta) = \cos(\dfrac{\pi}{2} - \theta)$

For example:

  • In degrees, $\sin(80^\circ) = \cos(10^\circ)$
  • In radians: $\sin \Big(\dfrac{\pi}{8} \Big) = \cos \Big( \dfrac{3\pi}{8} \Big)$

Cotangent

The cotangent (cot) of an angle is defined as the reciprocal of the tangent.

Or: $\cot(\theta) = \dfrac{1}{\tan(\theta)}$

The tangent of an angle is equal to the cotangent of its complement

  • $\tan(\theta) = \cot(90^\circ - \theta) = \dfrac{1}{\tan(90^\circ - \theta)}$
  • $\tan(\theta) = \cot \Big( \dfrac{\pi}{2}-\theta \Big)$

For example:

  • In degrees, $\tan(18^\circ) = \cot(72^\circ)$
  • In radians: $\tan \Big(\dfrac{\pi}{10} \Big) = \cot \Big( \dfrac{4\pi}{10} \Big)$

In Other Quadrants

The same relationship holds in the other quadrants, but watch for the negatives.

In all of these examples,

1st Quadrant:

  • $(90^\circ - \theta) + (\theta) = 90^\circ \; \; \textit{so}$
  • $\sin(90^\circ - \theta) = \cos(\theta)$

2nd Quadrant:

  • $(90^\circ + \theta) + (-\theta) = 90^\circ \; \; \textit{so}$
  • $\sin(90^\circ + \theta) = \cos(-\theta) = \cos(\theta)$

3rd Quadrant:

  • $(270^\circ - \theta) + (-180^\circ + \theta) = 90^\circ \; \; \textit{so}$
  • $\sin(270^\circ - \theta) = \cos(-180^\circ + \theta) = -\cos(\theta)$

4th Quadrant:

  • $(270^\circ + \theta) + (-180^\circ - \theta) = 90^\circ \; \; \textit{so}$
  • $\sin(270^\circ + \theta) = \cos(-180^\circ - \theta) = -\cos(\theta)$

Notice that the calculations for cos give us the correct sign for sin in that quadrant.

We can do similar calculations for cos and tan. The results are summarised in this table:

01.5tblcompvals.gif

Notice that adding/subtracting the angles in each entry gives either $\dfrac{\pi}{2} \text{ or } \dfrac{3\pi}{2}$.

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