6.03 Symmetries
Symmetries
There are many symmetries in the functions for sine and cosine from the unit circle.
Comparing the coordinates of the points around the circle leads to the following rules:
If we focus on the $y$y-coordinate, we can find:
| $\sin\left(180^\circ-\theta\right)$sin(180°−θ) | $=$= | $\sin\theta$sinθ |
| $\sin\left(180^\circ+\theta\right)$sin(180°+θ) | $=$= | $-\sin\theta$−sinθ |
| $\sin\left(-\theta\right)$sin(−θ) | $=$= | $-\sin\theta$−sinθ |
If we focus on the $x$x-coordinate, we can find:
| $\cos\left(180^\circ-\theta\right)$cos(180°−θ) | $=$= | $-\cos\theta$−cosθ |
| $\cos\left(180^\circ+\theta\right)$cos(180°+θ) | $=$= | $-\cos\theta$−cosθ |
| $\cos\left(-\theta\right)$cos(−θ) | $=$= | $\cos\theta$cosθ |
By using the definition $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ similar rules can be created for $\tan\theta$tanθ.
Further symmetries
Let's look at the relationship between trigonometric functions of complementary angles, that is, angles that add to $90^\circ$90°.
From the diagram, notice that the triangle formed by the angle $\theta$θ to the point $P$P has its sides flipped to form the triangle at angle $90^\circ-\theta$90°−θ to point $Q$Q. Since the $x$x- and $y$y-coordinates have swapped when comparing the two points, the following rules can be extracted:
| $\sin\left(90^\circ-\theta\right)$sin(90°−θ) | $=$= | $\cos\theta$cosθ |
| $\cos\left(90^\circ-\theta\right)$cos(90°−θ) | $=$= | $\sin\theta$sinθ |
Practice questions
Question 1
Given that $\sin x=0.46$sinx=0.46, find the exact value of $\cos\left(90^\circ-x\right)$cos(90°−x).
Question 2
Prove that $\frac{\sin x\sin\left(90^\circ-x\right)}{\cos x\cos\left(90^\circ-x\right)}=1$sinxsin(90°−x)cosxcos(90°−x)=1