Special Triangles, Unit Circle in Trigonometry - Precalculus

1. What are the side ratios in a $\frac{\pi}{4}$-$\frac{\pi}{4}$-$\frac{\pi}{2}$ triangle?

   $1:1:\sqrt{2}$. 45-45-90 triangle has equal legs and hypotenuse = leg × $\sqrt{2}$.

2. What identity expresses $\cos(2\pi-x)$ in terms of $\cos(x)$?

   $\cos(2\pi-x)=\cos(x)$. Clockwise rotation by $x$ from $2\pi$ keeps x-coordinate (cosine) same.

3. What is $\sin(2\pi-x)$ in terms of $\sin(x)$?

   $\sin(2\pi-x)=-\sin(x)$. Angle $2\pi-x$ is in quadrant IV, where sine is negative.

4. What identity expresses $\sin(\pi-x)$ in terms of $\sin(x)$?

   $\sin(\pi-x)=\sin(x)$. Reflection across y-axis keeps y-coordinate (sine) unchanged.

5. What are $\sin(\pi\/6)$, $\cos(\pi\/6)$, and $\tan(\pi\/6)$?

   $\sin(\pi/6)=\frac{1}{2}$, $\cos(\pi/6)=\frac{\sqrt{3}}{2}$, $\tan(\pi/6)=\frac{\sqrt{3}}{3}$. From 30-60-90 triangle: opposite/hyp = 1/2, adjacent/hyp = $\sqrt{3}$/2.

6. What is $\sin\left(\frac{\pi}{3}\right)$?

   $\frac{\sqrt{3}}{2}$. At $60°$, sine equals opposite/hypotenuse = $\sqrt{3}/2$ in the 30-60-90 triangle.

7. What identity expresses $\tan(\pi-x)$ in terms of $\tan(x)$?

   $\tan(\pi-x)=-\tan(x)$. Since $\tan = \sin/\cos$ and cosine changes sign, tangent negates.

8. What is $\sin(\pi-x)$ in terms of $\sin(x)$?

   $\sin(\pi-x)=\sin(x)$. Supplementary angles have the same sine value in quadrants I and II.

9. What is $\sin\left(\frac{\pi}{6}\right)$?

   $\frac{1}{2}$. At $30°$, sine equals opposite/hypotenuse = $1/2$ in the 30-60-90 triangle.

10. What is the unit-circle coordinate for angle $\frac{\pi}{4}$?

    $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$. Unit circle point at $45°$ has coordinates $(\cos 45°, \sin 45°)$.

11. What are $\sin(\pi\/4)$, $\cos(\pi\/4)$, and $\tan(\pi\/4)$?

    $\sin(\pi/4)=\frac{\sqrt{2}}{2}$, $\cos(\pi/4)=\frac{\sqrt{2}}{2}$, $\tan(\pi/4)=1$. From 45-45-90 triangle: both legs equal, so sine = cosine = $\frac{1}{\sqrt{2}}$.

12. What are $\sin(\pi\/3)$, $\cos(\pi\/3)$, and $\tan(\pi\/3)$?

    $\sin(\pi/3)=\frac{\sqrt{3}}{2}$, $\cos(\pi/3)=\frac{1}{2}$, $\tan(\pi/3)=\sqrt{3}$. From 30-60-90 triangle: opposite/hyp = $\sqrt{3}$/2, adjacent/hyp = 1/2.

13. What identity expresses $\cos(\pi-x)$ in terms of $\cos(x)$?

    $\cos(\pi-x)=-\cos(x)$. Reflection across y-axis negates x-coordinate (cosine).

14. What is $\tan(2\pi-\pi\/4)$?

    $-1$. $2\pi - \pi/4 = 7\pi/4$; using $\tan(2\pi-x)=-\tan(x)$ gives $-\tan(\pi/4)=-1$.

15. What is $\cos\left(\frac{\pi}{4}\right)$?

    $\frac{\sqrt{2}}{2}$. At $45°$, cosine equals adjacent/hypotenuse = $1/\sqrt{2} = \sqrt{2}/2$.

16. What is $\cos\left(\frac{\pi}{6}\right)$?

    $\frac{\sqrt{3}}{2}$. At $30°$, cosine equals adjacent/hypotenuse = $\sqrt{3}/2$ in the 30-60-90 triangle.

17. What identity expresses $\tan(\pi+x)$ in terms of $\tan(x)$?

    $\tan(\pi+x)=\tan(x)$. Both sine and cosine negate, so their ratio remains unchanged.

18. What is $\cos(\pi+x)$ in terms of $\cos(x)$?

    $\cos(\pi+x)=-\cos(x)$. Adding $\pi$ reflects across origin, changing cosine's sign.

19. What is $\tan(\pi-x)$ in terms of $\tan(x)$?

    $\tan(\pi-x)=-\tan(x)$. Since $\sin$ stays same and $\cos$ changes sign, $\tan$ changes sign.

20. What is $\tan(2\pi-x)$ in terms of $\tan(x)$?

    $\tan(2\pi-x)=-\tan(x)$. Since $\sin$ is negative and $\cos$ is positive, $\tan$ is negative.

21. What is $\cos(\pi-x)$ in terms of $\cos(x)$?

    $\cos(\pi-x)=-\cos(x)$. Supplementary angles have opposite cosine values in quadrants I and II.

22. What is the unit-circle coordinate for angle $\frac{\pi}{3}$?

    $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. Unit circle point at $60°$ has coordinates $(\cos 60°, \sin 60°)$.

23. What is $\sin(2\pi-\pi\/3)$?

    $-\frac{\sqrt{3}}{2}$. $2\pi - \pi/3 = 5\pi/3$; using $\sin(2\pi-x)=-\sin(x)$ gives $-\sin(\pi/3)$.

24. What is the unit-circle definition of $\sin(x)$ and $\cos(x)$ using the point $P(x)$?

    $P(x)=(\cos(x),\sin(x))$. Point at angle $x$ on unit circle has coordinates (cos, sin).

25. What is $\tan\left(\frac{\pi}{3}\right)$?

    $\sqrt{3}$. At $60°$, tangent equals opposite/adjacent = $\sqrt{3}/1 = \sqrt{3}$.

26. What is $\tan\left(\frac{\pi}{4}\right)$?

    $1$. At $45°$, tangent equals opposite/adjacent = $1/1 = 1$.

27. What is $\cos(2\pi-x)$ in terms of $\cos(x)$?

    $\cos(2\pi-x)=\cos(x)$. Angle $2\pi-x$ is in quadrant IV, where cosine is positive.

28. What identity expresses $\tan(2\pi-x)$ in terms of $\tan(x)$?

    $\tan(2\pi-x)=-\tan(x)$. Since sine negates and cosine stays same, tangent negates.

29. What is $\sin\left(\frac{\pi}{4}\right)$?

    $\frac{\sqrt{2}}{2}$. At $45°$, sine equals opposite/hypotenuse = $1/\sqrt{2} = \sqrt{2}/2$.

30. What identity expresses $\sin(\pi+x)$ in terms of $\sin(x)$?

    $\sin(\pi+x)=-\sin(x)$. Rotation by $\pi$ negates both coordinates, so sine becomes negative.

31. What are the side ratios of a $30^ -60^ -90^ $ triangle with shortest leg $1$?

    $1:\sqrt{3}:2$. In a 30-60-90 triangle, sides are in ratio short leg : long leg : hypotenuse.

32. What identity expresses $\sin(2\pi-x)$ in terms of $\sin(x)$?

    $\sin(2\pi-x)=-\sin(x)$. Clockwise rotation by $x$ from $2\pi$ negates y-coordinate (sine).

33. What is $\tan\left(\frac{\pi}{6}\right)$?

    $\frac{\sqrt{3}}{3}$. At $30°$, tangent equals opposite/adjacent = $1/\sqrt{3} = \sqrt{3}/3$.

34. What is $\sin(\pi+x)$ in terms of $\sin(x)$?

    $\sin(\pi+x)=-\sin(x)$. Adding $\pi$ reflects across origin, changing sine's sign.

35. What are the side ratios in a $\frac{\pi}{6}$-$\frac{\pi}{3}$-$\frac{\pi}{2}$ triangle?

    $1:\sqrt{3}:2$. 30-60-90 triangle has sides opposite to 30°:60°:90° in ratio 1:$\sqrt{3}$:2.

36. What is $\tan(\pi+x)$ in terms of $\tan(x)$?

    $\tan(\pi+x)=\tan(x)$. Since both $\sin$ and $\cos$ change sign, $\tan$ stays the same.

37. What is $\sin(\pi-\pi\/6)$?

    $\frac{1}{2}$. $\pi - \pi/6 = 5\pi/6$; using $\sin(\pi-x)=\sin(x)$ gives $\sin(\pi/6)=1/2$.

38. What identity expresses $\cos(\pi+x)$ in terms of $\cos(x)$?

    $\cos(\pi+x)=-\cos(x)$. Rotation by $\pi$ negates both coordinates, so cosine becomes negative.

39. What is $\cos\left(\frac{\pi}{3}\right)$?

    $\frac{1}{2}$. At $60°$, cosine equals adjacent/hypotenuse = $1/2$ in the 30-60-90 triangle.

40. What is the unit-circle coordinate for angle $\frac{\pi}{6}$?

    $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$. Unit circle point at $30°$ has coordinates $(\cos 30°, \sin 30°)$.

41. What are the side ratios of a $45^ -45^ -90^ $ triangle with legs $1$ and $1$?

    $1:1:\sqrt{2}$. In a 45-45-90 triangle, the hypotenuse equals leg times $\sqrt{2}$.

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