Trigonometry/Worked Example: Simplifying Angles

Worked Examples in Simplifying Angles

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Sign Changes

Sign changes (or otherwise)  $\displaystyle \cos( -3x )$ We know $\displaystyle \cos( -t ) = \cos( t )$ so $\displaystyle \cos( -3x ) = \cos( 3x )$

Sign changes (or otherwise)  $\displaystyle \sin( 180^\circ - \theta )$ We know $\displaystyle \sin( -t ) = -\sin( t )$ so: $\displaystyle \sin( 180^\circ - \theta ) = - \sin( \theta -180^\circ)$ We swapped the order of the terms at the same time just to save having to write $\displaystyle -180^\circ + \theta$, saving us one plus sign! Of course we can do that because the sum of two terms does not depend on their order. We also know that shifting the argument of sine (or cosine) by $\pm$ 180 degrees inverts the sign. So we can now remove the -180 and invert the sign to get: $\displaystyle \sin( 180^\circ - \theta ) = \sin \theta$

Sign changes (or otherwise)  $\displaystyle \cos( 360^\circ - t )$ We know shifting by 180 degrees inverts the sign. Shifting by 360 degrees is shifting by 180 degrees twice. Another way to think about it is that we have gone one complete revolution round the unit circle. Anyway, the 360 degrees in the expression makes no difference at all, so we have. $\displaystyle \cos( 360^\circ - t ) = \cos( -t)$ and we also know $\displaystyle \cos( -t ) = \cos( t )$ so $\displaystyle \cos( 360^\circ - t ) = \cos t$

Sign changes (or otherwise)  $\displaystyle \cos( -5x )\sin^2( 180^\circ - t )$ The minus in the $\displaystyle -5x$ will have no effect on the result since it is 'buried' inside the cosine. Likewise the 180 degree shift and the minus in the sine will have no effect on the sign of the result, since quite aside from the fact that they cancel each other, the sine is squared. (to spell that out, if we had got -sine of some expression, that all being squared would remove the negative sign again). So: $\displaystyle \cos( -5x )\sin^2( 180^\circ - t )= \cos5x\sin^2 t$
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Cosine to Sine

Complementary angles are pairs of angles that add up to $\displaystyle 90^\circ$ or if we are using Radian measure, $\displaystyle \frac{\pi}{2}$.

In a right angle triangles the other two angles, the two that are not the right angle, are complementary to each other. From the definitions of cosine and sine the cosine of an angle is the sine of the complementary angle. Also the sine of an angle is the cosine of the complementary angle.

Complementary Angles  Complementary angles: $\displaystyle \cos( 90^\circ - \theta ) = \sin \theta$ $\displaystyle \sin( 90^\circ - \theta ) = \cos \theta$

Cosine is an even function  Because cosine is an even function, $\displaystyle \cos( \theta - 90^\circ ) = \cos( 90^\circ-\theta )$ so $\displaystyle \cos( \theta - 90^\circ ) = \sin \theta$

Sine is an odd function  Because sine is an odd function, $\displaystyle \sin( \theta - 90^\circ ) = -\sin( 90^\circ-\theta )$ so $\displaystyle \sin( \theta - 90^\circ ) = - \cos \theta$

Adding 90o at a time  We can keep adding or subtracting 90o and switch between sine and cosine and possibly switch signs. We need to be careful to get the signs right. You can look at the graph to figure these ones out as needed, or just make sure you know about complementary angles, that sine is odd that cosine is even, and that adding or subtracting 180o flips the sign. $\displaystyle \sin( \theta - 180^\circ) = -\sin \theta$ add 180o flips the sign. $\displaystyle \sin( \theta - 90^\circ ) = -\cos \theta$ taking the negative, then complementary angle (one sign flip) $\displaystyle \sin \theta = \sin \theta$ $\displaystyle \sin( \theta + 90^\circ ) = \cos \theta$ subtract 180o, then negative, then complementary angle (two sign flips). $\displaystyle \sin( \theta + 180^\circ ) = -\sin \theta$ 180o flips the sign once. $\displaystyle \sin( \theta + 270^\circ ) = -\cos \theta$ subtract 360o, then negative, then complementary angle (three sign flips) $\displaystyle \sin( \theta + 360^\circ ) = \sin \theta$ 360o flips the sign twice and in these ones the step of taking the negative does not flip the sign since we are dealing with cosine $\displaystyle \cos( \theta - 180^\circ) = -\cos \theta$ add 180o flips the sign. $\displaystyle \cos( \theta - 90^\circ ) = \sin \theta$ taking the negative, then complementary angle (no sign flips) $\displaystyle \cos( \theta ) = \cos \theta$ $\displaystyle \cos( \theta + 90^\circ ) = -\sin \theta$ subtract 180o, then negative, then complementary angle (one sign flip). $\displaystyle \cos( \theta + 180^\circ ) = -\cos \theta$ 180o flips the sign once. $\displaystyle \cos( \theta + 270^\circ ) = \sin \theta$ subtract 360o, then negative, then complementary angle (two sign flips) $\displaystyle \cos( \theta + 360^\circ ) = \cos \theta$ 360o flips the sign twice

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Addition Formulas

Using Addition Formulas

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