Symmetry and Periodicity of Trigonometric Functions - Precalculus

1. What periodicity identity shows $\tan(\theta+\pi)$ in simplest form?

   $\tan(\theta+\pi)=\tan\theta$. Tangent has period $\pi$, so adding $\pi$ returns the same value.

2. What symmetry identity relates $\tan(-\theta)$ to $\tan\theta$?

   $\tan(-\theta)=-\tan\theta$. Tangent inherits odd symmetry from $\frac{\sin\theta}{\cos\theta}$.

3. What symmetry identity relates $\sin(-\theta)$ to $\sin\theta$?

   $\sin(-\theta)=-\sin\theta$. Sine is odd, so negative angles give opposite values.

4. What is the unit-circle symmetry identity for sine with a $\pi$ shift?

   $\sin(\theta+\pi)=-\sin\theta$. Adding $\pi$ rotates 180°, flipping the point through the origin.

5. What is the fundamental period of $\tan\theta$ on the unit circle?

   Period $=\pi$. Tangent repeats every half rotation since $\tan(\theta+\pi)=\tan\theta$.

6. What identity defines tangent in terms of unit circle coordinates $(x,y)$?

   $\tan\theta=\frac{y}{x}$. Tangent is the ratio of $y$ to $x$ coordinates on the unit circle.

7. What is $\tan(-\frac{\pi}{4})$ using odd symmetry?

   $-1$. Since $\tan(-\theta)=-\tan\theta$ and $\tan(\frac{\pi}{4})=1$.

8. What is the cofunction identity for cosine using a $\frac{\pi}{2}$ shift?

   $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$. The complement of an angle exchanges the roles of sine and cosine.

9. What symmetry identity relates $\cos(-\theta)$ to $\cos\theta$?

   $\cos(-\theta)=\cos\theta$. Cosine is even, so negative angles give the same value.

10. What is the symmetry identity for cotangent using even-odd properties?

    $\cot(-\theta)=-\cot\theta$. Cotangent is odd since $\cot\theta = \frac{\cos\theta}{\sin\theta}$ (even/odd = odd).

11. What unit-circle identity defines cosine and sine as coordinates at angle $\theta$?

    Point is $(\cos\theta,\sin\theta)$ on the unit circle. The unit circle has radius 1, so any point at angle $\theta$ has these coordinates.

12. What is the unit-circle symmetry identity for tangent with a $\pi$ shift?

    $\tan(\theta+\pi)=\tan\theta$. Since both sine and cosine flip signs with $\pi$, their ratio stays the same.

13. What is $\sin(-\frac{\pi}{6})$ using odd symmetry?

    $-\frac{1}{2}$. Since $\sin(-\theta)=-\sin\theta$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$.

14. What is the sine identity for reflecting across the $y$-axis: $\pi-\theta$?

    $\sin(\pi-\theta)=\sin\theta$. Points at $\theta$ and $\pi-\theta$ have the same y-coordinate (sine value).

15. What periodicity identity shows $\sin(\theta+2\pi)$ in simplest form?

    $\sin(\theta+2\pi)=\sin\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.

16. What is the sine identity for $2\pi-\theta$ using unit-circle symmetry?

    $\sin(2\pi-\theta)=-\sin\theta$. Going clockwise by $\theta$ from 0 gives the opposite y-coordinate.

17. What is the cosine identity for reflecting across the $y$-axis: $\pi-\theta$?

    $\cos(\pi-\theta)=-\cos\theta$. Points at $\theta$ and $\pi-\theta$ have opposite x-coordinates.

18. What is $\sin(\frac{11\pi}{6})$ using periodicity and unit circle values?

    $-\frac{1}{2}$. $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$, and $\sin(-\frac{\pi}{6})=-\frac{1}{2}$.

19. What is the even-odd symmetry identity for sine on the unit circle?

    $\sin(-\theta)=-\sin\theta$. Sine is odd: reflecting across the y-axis negates the y-coordinate.

20. What is the even-odd symmetry identity for tangent?

    $\tan(-\theta)=-\tan\theta$. Tangent is odd since $\tan\theta = \frac{\sin\theta}{\cos\theta}$ (odd/even = odd).

21. What is the even/odd classification of $\cos\theta$?

    $\cos\theta$ is even. Even functions satisfy $f(-x)=f(x)$; cosine has this property.

22. What is the symmetry identity for secant using even-odd properties?

    $\sec(-\theta)=\sec\theta$. Secant is even since $\sec\theta = \frac{1}{\cos\theta}$ and cosine is even.

23. What is the period of $\cos\theta$?

    $2\pi$. Cosine completes one full cycle every $2\pi$ radians.

24. What is the unit-circle symmetry identity for cosine with a $\pi$ shift?

    $\cos(\theta+\pi)=-\cos\theta$. A $\pi$ rotation negates both coordinates, changing the sign of cosine.

25. What is the even-odd symmetry identity for cosine on the unit circle?

    $\cos(-\theta)=\cos\theta$. Cosine is even: reflecting across the y-axis doesn't change the x-coordinate.

26. What is the cosine identity for $2\pi-\theta$ using unit-circle symmetry?

    $\cos(2\pi-\theta)=\cos\theta$. Going clockwise by $\theta$ from 0 gives the same x-coordinate.

27. What is the even/odd classification of $\tan\theta$?

    $\tan\theta$ is odd. Since $\tan(-\theta)=-\tan\theta$, tangent is an odd function.

28. What periodicity identity shows $\cos(\theta+2\pi)$ in simplest form?

    $\cos(\theta+2\pi)=\cos\theta$. Adding $2\pi$ completes one full rotation, returning to the same value.

29. What is the fundamental period of $\sin\theta$ on the unit circle?

    Period $=2\pi$. One full rotation around the unit circle returns to the same point.

30. What is the symmetry identity for cosecant using even-odd properties?

    $\csc(-\theta)=-\csc\theta$. Cosecant is odd since $\csc\theta = \frac{1}{\sin\theta}$ and sine is odd.

31. What is the even/odd classification of $\sin\theta$?

    $\sin\theta$ is odd. Odd functions satisfy $f(-x)=-f(x)$; sine has this property.

32. What is the periodicity identity for sine for any integer $k$?

    $\sin(\theta+2\pi k)=\sin\theta$. Adding any multiple of $2\pi$ completes full rotations, returning to the same point.

33. What is the fundamental period of $\cos\theta$ on the unit circle?

    Period $=2\pi$. Cosine repeats every full rotation ($2\pi$ radians) around the circle.

34. What is the period of $\sin\theta$?

    $2\pi$. Sine completes one full cycle every $2\pi$ radians.

35. What is the period of $\tan\theta$?

    $\pi$. Tangent repeats every $\pi$ radians, half the period of sine/cosine.

36. What is the cofunction identity for sine using a $\frac{\pi}{2}$ shift?

    $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$. Complementary angles swap sine and cosine values on the unit circle.

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