Solving Right Triangles: Pythagorean Theorem, Trigonometry - Geometry

1. What is the complementary-angle identity that relates $\sin(\theta)$ to a cosine expression?

   $\sin(\theta)=\cos(90^\circ-\theta)$. Sine of an angle equals cosine of its complement.

2. What does it mean for angles $\alpha$ and $\beta$ to be complementary in degrees?

   $\alpha+\beta=90^\circ$. Two angles are complementary when they sum to $90^\circ$.

3. What is $\cos(60^\circ)$ written as a sine of a complementary angle?

   $\cos(60^\circ)=\sin(30^\circ)$. Since $60^\circ + 30^\circ = 90^\circ$, they are complementary.

4. What does it mean for angles $\alpha$ and $\beta$ to be complementary in radians?

   $\alpha+\beta=\frac{\pi}{2}$. Two angles are complementary when they sum to $\frac{\pi}{2}$.

5. Find and correct the identity error: $\sin(\theta)=\sin(90^\circ-\theta)$.

   Correct: $\sin(\theta)=\cos(90^\circ-\theta)$. Sine equals cosine of complement, not sine of complement.

6. What is $\sin\left(\frac{\pi}{6}\right)$ rewritten as a cosine of a complementary angle?

   $\sin\left(\frac{\pi}{6}\right)=\cos\left(\frac{\pi}{3}\right)$. Since $\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}$, they are complementary.

7. Find the complement: If $\theta=11^\circ$, what is $90^\circ-\theta$?

   $79^\circ$. Complementary angles sum to $90^\circ$.

8. Find the complement: If $\theta=58^\circ$, what is $90^\circ-\theta$?

   $32^\circ$. Complementary angles sum to $90^\circ$.

9. What is $\sin(45^\circ)$ written as a cosine of a complementary angle?

   $\sin(45^\circ)=\cos(45^\circ)$. $45^\circ$ is its own complement since $45^\circ + 45^\circ = 90^\circ$.

10. What is $\sin(90^\circ-\theta)$ equal to in terms of $\cos(\theta)$?

    $\sin(90^\circ-\theta)=\cos(\theta)$. Sine of $90^\circ-\theta$ is the cosine of $\theta$.

11. If $\sin(\theta)=\cos(35^\circ)$ and $\theta$ is acute, what is $\theta$?

    $\theta=55^\circ$. Since $\sin(\theta) = \cos(35^\circ)$, then $\theta = 90^\circ - 35^\circ$.

12. Which statement is always true for complementary angles $\alpha$ and $\beta$: $\sin(\alpha)=\cos(\beta)$ or $\sin(\alpha)=\sin(\beta)$?

    $\sin(\alpha)=\cos(\beta)$. Complementary angles have equal sine and cosine cofunctions.

13. What is $\sin(60^\circ)$ written as a cosine of a complementary angle?

    $\sin(60^\circ)=\cos(30^\circ)$. Since $60^\circ + 30^\circ = 90^\circ$, they are complementary.

14. What is $\cos\left(\frac{\pi}{6}\right)$ rewritten as a sine of a complementary angle?

    $\cos\left(\frac{\pi}{6}\right)=\sin\left(\frac{\pi}{3}\right)$. Since $\frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2}$, they are complementary.

15. What is $\sin(75^\circ)$ rewritten using cosine of a complementary angle?

    $\sin(75^\circ)=\cos(15^\circ)$. Since $75^\circ + 15^\circ = 90^\circ$, they are complementary.

16. In a right triangle, if one acute angle is $\theta$, what is the other acute angle in radians?

    $\frac{\pi}{2}-\theta$. Right triangles have two acute angles that sum to $\frac{\pi}{2}$.

17. What is $\cos(75^\circ)$ rewritten using sine of a complementary angle?

    $\cos(75^\circ)=\sin(15^\circ)$. Since $75^\circ + 15^\circ = 90^\circ$, they are complementary.

18. What is $\sin\left(\frac{\pi}{12}\right)$ rewritten using cosine of a complementary angle?

    $\sin\left(\frac{\pi}{12}\right)=\cos\left(\frac{5\pi}{12}\right)$. Since $\frac{\pi}{12} + \frac{5\pi}{12} = \frac{\pi}{2}$, they are complementary.

19. Identify the correct value: What is $\sin(30^\circ)$ written as a cosine of a complementary angle?

    $\sin(30^\circ)=\cos(60^\circ)$. Since $30^\circ + 60^\circ = 90^\circ$, they are complementary.

20. If $\cos(\theta)=\sin(18^\circ)$ and $\theta$ is acute, what is $\theta$?

    $\theta=72^\circ$. Since $\cos(\theta) = \sin(18^\circ)$, then $\theta = 90^\circ - 18^\circ$.

21. What is $\cos(0^\circ)$ rewritten as a sine of a complementary angle?

    $\cos(0^\circ)=\sin(90^\circ)$. Since $0^\circ + 90^\circ = 90^\circ$, they are complementary.

22. What is $\cos(45^\circ)$ written as a sine of a complementary angle?

    $\cos(45^\circ)=\sin(45^\circ)$. $45^\circ$ is its own complement since $45^\circ + 45^\circ = 90^\circ$.

23. What is $\cos\left(\frac{\pi}{2}-\theta\right)$ equal to in terms of $\sin(\theta)$?

    $\cos\left(\frac{\pi}{2}-\theta\right)=\sin(\theta)$. Cosine of complement equals sine of original angle.

24. If $\alpha$ and $\beta$ are complementary and $\cos(\alpha)=\frac{7}{9}$, what is $\sin(\beta)$?

    $\sin(\beta)=\frac{7}{9}$. For complementary angles, $\cos(\alpha) = \sin(\beta)$.

25. Find the complement: If $\theta=23^\circ$, what is $90^\circ-\theta$?

    $67^\circ$. Complementary angles sum to $90^\circ$.

26. In a right triangle, which function of $\theta$ equals the ratio opposite over hypotenuse?

    $\sin(\theta)$. Sine is defined as opposite over hypotenuse.

27. Identify the correct value: What is $\cos(30^\circ)$ written as a sine of a complementary angle?

    $\cos(30^\circ)=\sin(60^\circ)$. Since $30^\circ + 60^\circ = 90^\circ$, they are complementary.

28. If $\alpha$ and $\beta$ are complementary and $\sin(\beta)=0.4$, what is $\cos(\alpha)$?

    $\cos(\alpha)=0.4$. For complementary angles, $\sin(\beta) = \cos(\alpha)$.

29. If $\theta$ is acute and $\sin(\theta)=\frac{12}{13}$, what is $\cos(90^\circ-\theta)$?

    $\cos(90^\circ-\theta)=\frac{12}{13}$. Using the cofunction identity for complementary angles.

30. If $\alpha$ and $\beta$ are complementary and $\sin(\alpha)=\frac{3}{5}$, what is $\cos(\beta)$?

    $\cos(\beta)=\frac{3}{5}$. For complementary angles, $\sin(\alpha) = \cos(\beta)$.

31. What is $\cos(90^\circ)$ rewritten as a sine of a complementary angle?

    $\cos(90^\circ)=\sin(0^\circ)$. Since $90^\circ + 0^\circ = 90^\circ$, they are complementary.

32. In a right triangle, which function of $\theta$ equals the ratio adjacent over hypotenuse?

    $\cos(\theta)$. Cosine is defined as adjacent over hypotenuse.

33. What is $\sin(0^\circ)$ rewritten as a cosine of a complementary angle?

    $\sin(0^\circ)=\cos(90^\circ)$. Since $0^\circ + 90^\circ = 90^\circ$, they are complementary.

34. What is the identity for $\sin(\theta)$ using radians and complementary angles?

    $\sin(\theta)=\cos\left(\frac{\pi}{2}-\theta\right)$. Radian form: sine equals cosine of complement.

35. Find the complement: If $\theta=\frac{\pi}{8}$, what is $\frac{\pi}{2}-\theta$?

    $\frac{3\pi}{8}$. Complementary angles sum to $\frac{\pi}{2}$.

36. In a right triangle, if the acute angles are $\theta$ and $90^\circ-\theta$, what is $\sin(\theta)$ equal to?

    $\cos(90^\circ-\theta)$. Using the cofunction identity in right triangles.

37. Find the complement: If $\theta=\frac{\pi}{3}$, what is $\frac{\pi}{2}-\theta$?

    $\frac{\pi}{6}$. Complementary angles sum to $\frac{\pi}{2}$.

38. What is $\cos(10^\circ)$ rewritten using sine of a complementary angle?

    $\cos(10^\circ)=\sin(80^\circ)$. Since $10^\circ + 80^\circ = 90^\circ$, they are complementary.

39. In a right triangle, if one acute angle is $\theta$, what is the other acute angle in degrees?

    $90^\circ-\theta$. Right triangles have two acute angles that sum to $90^\circ$.

40. What is $\sin(90^\circ)$ rewritten as a cosine of a complementary angle?

    $\sin(90^\circ)=\cos(0^\circ)$. Since $90^\circ + 0^\circ = 90^\circ$, they are complementary.

41. What is $\cos\left(\frac{\pi}{12}\right)$ rewritten using sine of a complementary angle?

    $\cos\left(\frac{\pi}{12}\right)=\sin\left(\frac{5\pi}{12}\right)$. Since $\frac{\pi}{12} + \frac{5\pi}{12} = \frac{\pi}{2}$, they are complementary.

42. In a right triangle, if the acute angles are $\theta$ and $90^\circ-\theta$, what is $\cos(\theta)$ equal to?

    $\sin(90^\circ-\theta)$. Using the cofunction identity in right triangles.

43. What is $\cos(90^\circ-\theta)$ equal to in terms of $\sin(\theta)$?

    $\cos(90^\circ-\theta)=\sin(\theta)$. Cosine of $90^\circ-\theta$ is the sine of $\theta$.

44. If $\alpha$ and $\beta$ are complementary and $\cos(\beta)=0.92$, what is $\sin(\alpha)$?

    $\sin(\alpha)=0.92$. For complementary angles, $\cos(\beta) = \sin(\alpha)$.

45. What is $\sin\left(\frac{\pi}{2}-\theta\right)$ equal to in terms of $\cos(\theta)$?

    $\sin\left(\frac{\pi}{2}-\theta\right)=\cos(\theta)$. Sine of complement equals cosine of original angle.

46. If $\sin(\theta)=\cos(\theta)$ and $\theta$ is acute, what is $\theta$?

    $\theta=45^\circ$. Only at $45^\circ$ are sine and cosine equal.

47. What is the identity for $\cos(\theta)$ using radians and complementary angles?

    $\cos(\theta)=\sin\left(\frac{\pi}{2}-\theta\right)$. Radian form: cosine equals sine of complement.

48. If $\theta$ is acute and $\cos(\theta)=\frac{5}{12}$, what is $\sin(90^\circ-\theta)$?

    $\sin(90^\circ-\theta)=\frac{5}{12}$. Using the cofunction identity for complementary angles.

49. If $\sin(\alpha)=\cos(\beta)$ and both angles are acute, what must be true about $\alpha$ and $\beta$?

    $\alpha+\beta=90^\circ$. The cofunction identity requires complementary angles.

50. Which statement is always true for complementary angles $\alpha$ and $\beta$: $\cos(\alpha)=\sin(\beta)$ or $\cos(\alpha)=\cos(\beta)$?

    $\cos(\alpha)=\sin(\beta)$. Complementary angles have equal cosine and sine cofunctions.

51. What is the complementary-angle identity that relates $\cos(\theta)$ to a sine expression?

    $\cos(\theta)=\sin(90^\circ-\theta)$. Cosine of an angle equals sine of its complement.

52. What is $\sin(10^\circ)$ rewritten using cosine of a complementary angle?

    $\sin(10^\circ)=\cos(80^\circ)$. Since $10^\circ + 80^\circ = 90^\circ$, they are complementary.

53. Find and correct the identity error: $\cos(\theta)=\cos(90^\circ-\theta)$.

    Correct: $\cos(\theta)=\sin(90^\circ-\theta)$. Cosine equals sine of complement, not cosine of complement.

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