Proving the Pythagorean Identity - Geometry

1. Find $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.

   $\sin(\theta)=\frac{3}{5}$. From $\tan(\theta)=\frac{3}{4}$, use identity $1+\tan^2(\theta)=\sec^2(\theta)$ to find sine.

2. Identify the correct value: if $\cos(\theta)=-\frac{\sqrt{3}}{2}$ in Quadrant III, what is $\sin(\theta)$?

   $\sin(\theta)=-\frac{1}{2}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{3}{4}=\frac{1}{4}$, negative in Q3.

3. Identify the right-triangle relation used to prove $\sin^2(\theta)+\cos^2(\theta)=1$.

   $a^2+b^2=c^2$. The Pythagorean theorem for right triangles with hypotenuse $c$.

4. What is $\sin(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant II?

   $\sin(\theta)=\frac{15}{17}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{64}{289}=\frac{225}{289}$, sine is positive in Q2.

5. What is $\sin(\theta)$ if $\cos(\theta)=\frac{12}{13}$ and $\theta$ is in Quadrant IV?

   $\sin(\theta)=-\frac{5}{13}$. Same calculation as Q1, but sine is negative in Quadrant IV.

6. What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I?

   $\cos(\theta)=\frac{4}{5}$. Using $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$, so $\cos(\theta)=\frac{4}{5}$ in Q1.

7. What is $\cos(\theta)$ if $\sin(\theta)=-\frac{5}{13}$ and $\theta$ is in Quadrant IV?

   $\cos(\theta)=\frac{12}{13}$. Same sine calculation, but cosine is positive in Quadrant IV.

8. State the definitions $\sin(\theta)$ and $\cos(\theta)$ in a right triangle.

   $\sin(\theta)=\frac{\text{opp}}{\text{hyp}},\;\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$. Standard trigonometric ratios in a right triangle.

9. What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant I?

   $\sin(\theta)>0$ and $\cos(\theta)>0$. In Quadrant I, both coordinates are positive.

10. What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II?

    $\cos(\theta)=-\frac{4}{5}$. Same calculation as Q1, but cosine is negative in Quadrant II.

11. What is the sign of $\tan(\theta)$ in Quadrant II?

    $\tan(\theta)<0$. In Quadrant II, $\sin>0$ and $\cos<0$, so $\tan<0$.

12. Find $\cos(\theta)$ if $\tan(\theta)=-\frac{5}{12}$ and $\theta$ is in Quadrant IV.

    $\cos(\theta)=\frac{12}{13}$. From $\tan(\theta)=-\frac{5}{12}$, find cosine using $1+\tan^2(\theta)=\sec^2(\theta)$.

13. Identify the expressions $\frac{a}{c}$ and $\frac{b}{c}$ in terms of $\theta$ for the Pythagorean identity proof.

    $\frac{a}{c}=\sin(\theta)$ and $\frac{b}{c}=\cos(\theta)$. These ratios equal sine and cosine by definition.

14. State the Pythagorean identity involving $\cot(\theta)$ and $\csc(\theta)$.

    $1+\cot^2(\theta)=\csc^2(\theta)$. Rearranged form of $\cot^2(\theta)=\csc^2(\theta)-1$.

15. State the quotient identity defining $\cot(\theta)$ in terms of $\cos(\theta)$ and $\sin(\theta)$.

    $\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}$. The basic quotient identity defining cotangent.

16. State the quotient identity defining $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$.

    $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. The basic quotient identity defining tangent.

17. Find $\sin(\theta)$ if $\cos(\theta)=\frac{24}{25}$ and $\theta$ is in Quadrant IV.

    $\sin(\theta)=-\frac{7}{25}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{576}{625}=\frac{49}{625}$, negative in Q4.

18. Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I.

    $\tan(\theta)=\frac{3}{4}$. Find $\cos(\theta)=\frac{4}{5}$ first, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

19. What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant III?

    $\sin(\theta)<0$ and $\cos(\theta)<0$. In Quadrant III, both coordinates are negative.

20. What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant IV?

    $\sin(\theta)<0$ and $\cos(\theta)>0$. In Quadrant IV, $x>0$ but $y<0$.

21. Find $\tan(\theta)$ if $\cos(\theta)=\frac{12}{13}$ and $\theta$ is in Quadrant IV.

    $\tan(\theta)=-\frac{5}{12}$. Find $\sin(\theta)=-\frac{5}{13}$ first, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

22. What is the unit-circle reason that $\sin^2(\theta)+\cos^2(\theta)=1$ is true?

    It is $x^2+y^2=1$ with $x=\cos(\theta)$, $y=\sin(\theta)$. Any point on the unit circle satisfies $x^2+y^2=1$.

23. Find $\sin(\theta)$ if $\tan(\theta)=-\frac{5}{12}$ and $\theta$ is in Quadrant IV.

    $\sin(\theta)=-\frac{5}{13}$. From $\tan(\theta)=-\frac{5}{12}$, find sine using $1+\tan^2(\theta)=\sec^2(\theta)$.

24. Identify the correct value: if $\sin(\theta)=\frac{1}{2}$ in Quadrant II, what is $\cos(\theta)$?

    $\cos(\theta)=-\frac{\sqrt{3}}{2}$. Using $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{1}{4}=\frac{3}{4}$, negative in Q2.

25. Choose the correct conclusion: if $\sin(\theta)=0$, what must $\cos(\theta)$ equal?

    $\cos(\theta)=\pm 1$. When sine is zero, the Pythagorean identity gives $\cos^2(\theta)=1$.

26. State the reciprocal identities for $\sec(\theta)$ and $\csc(\theta)$.

    $\sec(\theta)=\frac{1}{\cos(\theta)},\;\csc(\theta)=\frac{1}{\sin(\theta)}$. Standard reciprocal relationships for secant and cosecant.

27. What is the sign of $\tan(\theta)$ in Quadrant III?

    $\tan(\theta)>0$. In Quadrant III, both sine and cosine are negative, so $\tan>0$.

28. State the identity for $\tan^2(\theta)$ in terms of $\sec^2(\theta)$.

    $\tan^2(\theta)=\sec^2(\theta)-1$. Derived by dividing the primary Pythagorean identity by $\cos^2(\theta)$.

29. Find $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant III.

    $\sin(\theta)=-\frac{3}{5}$. Same calculation as Q1, but sine is negative in Quadrant III.

30. What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant II?

    $\sin(\theta)>0$ and $\cos(\theta)<0$. In Quadrant II, $y>0$ but $x<0$.

31. Find $\tan(\theta)$ if $\sin(\theta)=-\frac{1}{2}$ and $\cos(\theta)=\frac{\sqrt{3}}{2}$.

    $\tan(\theta)=-\frac{\sqrt{3}}{3}$. Using $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$.

32. What is the sign of $\tan(\theta)$ in Quadrant IV?

    $\tan(\theta)<0$. In Quadrant IV, $\sin<0$ and $\cos>0$, so $\tan<0$.

33. Identify the general method to find $\sin(\theta)$ given $\cos(\theta)$ and the quadrant.

    $\sin(\theta)=\pm\sqrt{1-\cos^2(\theta)}$ with sign from quadrant. Solve for sine using the identity, then apply the quadrant sign.

34. Find $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.

    $\cos(\theta)=\frac{4}{5}$. From $\tan(\theta)=\frac{3}{4}$, use identity $1+\tan^2(\theta)=\sec^2(\theta)$ to find cosine.

35. State the Pythagorean identity relating $\sin^2(\theta)$ and $\cos^2(\theta)$.

    $\sin^2(\theta)+\cos^2(\theta)=1$. This is the fundamental Pythagorean identity derived from the unit circle.

36. Find $\tan(\theta)$ if $\sin(\theta)=-\frac{7}{25}$ and $\theta$ is in Quadrant III.

    $\tan(\theta)=\frac{7}{24}$. Find $\cos(\theta)=-\frac{24}{25}$ first, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

37. Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II.

    $\tan(\theta)=-\frac{3}{4}$. Find $\cos(\theta)=-\frac{4}{5}$ first, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

38. What is $\sin(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant III?

    $\sin(\theta)=-\frac{15}{17}$. Same calculation as Q2, but sine is negative in Quadrant III.

39. Find $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant III.

    $\cos(\theta)=-\frac{4}{5}$. Same calculation as Q1, but cosine is negative in Quadrant III.

40. What is $\sin(\theta)$ if $\cos(\theta)=\frac{12}{13}$ and $\theta$ is in Quadrant I?

    $\sin(\theta)=\frac{5}{13}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{144}{169}=\frac{25}{169}$, so $\sin(\theta)=\frac{5}{13}$ in Q1.

41. Find $\sin(\theta)$ if $\cos(\theta)=-\frac{4}{5}$ and $\theta$ is in Quadrant II.

    $\sin(\theta)=\frac{3}{5}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{16}{25}=\frac{9}{25}$, positive in Q2.

42. What is $\cos(\theta)$ if $\sin(\theta)=-\frac{5}{13}$ and $\theta$ is in Quadrant III?

    $\cos(\theta)=-\frac{12}{13}$. Using $\sin^2(\theta)=1-\cos^2(\theta)=\frac{25}{169}$, cosine is negative in Q3.

43. Find $\cos(\theta)$ if $\sin(\theta)=\frac{7}{25}$ and $\theta$ is in Quadrant I.

    $\cos(\theta)=\frac{24}{25}$. Using $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{49}{625}=\frac{576}{625}$, positive in Q1.

44. What identity results after dividing $a^2+b^2=c^2$ by $c^2$ in a right triangle?

    $\left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\right)^2=1$. Dividing the Pythagorean theorem by $c^2$ normalizes to unit circle form.

45. Find $\tan(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant III.

    $\tan(\theta)=\frac{15}{8}$. Find $\sin(\theta)=-\frac{15}{17}$ first, then $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

46. Find $\tan(\theta)$ if $\cos(\theta)=-\frac{4}{5}$ and $\theta$ is in Quadrant II.

    $\tan(\theta)=-\frac{3}{4}$. Using known values: $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{\frac{3}{5}}{-\frac{4}{5}}$.

47. State the identity for $\cot^2(\theta)$ in terms of $\csc^2(\theta)$.

    $\cot^2(\theta)=\csc^2(\theta)-1$. Derived by dividing the primary Pythagorean identity by $\sin^2(\theta)$.

48. Identify the error: concluding $\cos(\theta)=\sqrt{1-\sin^2(\theta)}$ without using the quadrant.

    The sign is missing; $\cos(\theta)=\pm\sqrt{1-\sin^2(\theta)}$. The quadrant determines the sign, so both positive and negative are possible.

49. State the Pythagorean identity involving $\tan(\theta)$ and $\sec(\theta)$.

    $1+\tan^2(\theta)=\sec^2(\theta)$. Rearranged form of $\tan^2(\theta)=\sec^2(\theta)-1$.

50. Find $\cos(\theta)$ if $\sin(\theta)=\frac{7}{25}$ and $\theta$ is in Quadrant II.

    $\cos(\theta)=-\frac{24}{25}$. Same calculation as Q1, but cosine is negative in Quadrant II.

51. Identify the general method to find $\cos(\theta)$ given $\sin(\theta)$ and the quadrant.

    $\cos(\theta)=\pm\sqrt{1-\sin^2(\theta)}$ with sign from quadrant. Solve for cosine using the identity, then apply the quadrant sign.

52. Which substitution into $a^2+b^2=c^2$ leads directly to $\sin^2(\theta)+\cos^2(\theta)=1$?

    Use $a=c\sin(\theta)$ and $b=c\cos(\theta)$. Express the sides $a$ and $b$ in terms of hypotenuse $c$ and angle $\theta$.

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