Proving the Pythagorean Identity - Precalculus

1. What identity relates $\sin^2(\theta)$ and $\cos^2(\theta)$ for any angle $\theta$?

   $\sin^2(\theta)+\cos^2(\theta)=1$. Fundamental trig identity that holds for all angles.

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

   $\cos(\theta)=\frac{4}{5}$. Use $\cos^2=1-\sin^2=1-\frac{9}{25}=\frac{16}{25}$; QI means positive.

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

   $\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; positive in Q1.

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

   $-\frac{15}{8}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{15/17}{-8/17}=-\frac{15}{8}$.

5. What is the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$?

   $\sin^2(\theta)+\cos^2(\theta)=1$. Fundamental identity derived from the unit circle equation $x^2+y^2=1$.

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

   $\tan(\theta)>0$. Since both $\sin<0$ and $\cos<0$ in QIII, their ratio is positive.

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

   $\sin(\theta)>0,\ \cos(\theta)<0$. In Q2, $y$-values are positive and $x$-values are negative.

8. What is the sign of $\cos(\theta)$ in Quadrant II?

   $\cos(\theta)<0$. In QII, $x$-values are negative on the unit circle.

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

   $-\frac{3}{5}$. From $\tan(\theta)=\frac{3}{4}$ and Q3 signs, $\sin(\theta)=-\frac{3}{5}$.

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

    $\sin(\theta)>0,\ \cos(\theta)>0$. Both coordinates are positive in the first quadrant.

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

    $\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; positive in Q1.

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

    $\frac{4}{5}$. From $\tan(\theta)=\frac{3}{4}$ and $\sin^2+\cos^2=1$ with Q1 signs.

13. Find $\sin(\theta)$ if $\cos(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant II.

    $\sin(\theta)=\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QII means positive.

14. What is $\tan(\theta)$ if $\sin(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant III?

    $\frac{12}{5}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{-12/13}{-5/13}=\frac{12}{5}$.

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

    $\tan(\theta)<0$. Since $\sin>0$ and $\cos<0$ in QII, their ratio is negative.

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

    $\sin(\theta)<0,\ \cos(\theta)<0$. Both coordinates are negative in the third quadrant.

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

    $\cos(\theta)=-\frac{4}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII cos negative.

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

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

19. What is the sign of $\sin(\theta)$ in Quadrant III?

    $\sin(\theta)<0$. In QIII, $y$-values are negative on the unit circle.

20. What is the sign of $\sin(\theta)$ in Quadrant II?

    $\sin(\theta)>0$. In QII, $y$-values are positive on the unit circle.

21. What is the unit-circle justification for $\sin^2(\theta)+\cos^2(\theta)=1$ using a point $(x,y)$?

    On $x^2+y^2=1$, $x=\cos(\theta)$, $y=\sin(\theta)$. Points on unit circle satisfy $x^2+y^2=1$, giving the identity.

22. What is $\cos^2(\theta)$ rewritten using $\sin(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?

    $\cos^2(\theta)=1-\sin^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\sin^2(\theta)$ from both sides.

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

    $\sin(\theta)<0,\ \cos(\theta)>0$. In Q4, $x$-values are positive and $y$-values are negative.

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

    $-\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; negative in Q4.

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

    $\sin(\theta)=-\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QIV means negative.

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

    $\cos(\theta)=\frac{4}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.

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

    $-\frac{3}{4}$. Find $\cos(\theta)=-\frac{4}{5}$ in Q2, then $\tan(\theta)=\frac{3/5}{-4/5}=-\frac{3}{4}$.

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

    $\frac{3}{4}$. Find $\cos(\theta)=\frac{4}{5}$ in Q1, then $\tan(\theta)=\frac{3/5}{4/5}=\frac{3}{4}$.

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

    $\sin(\theta)=\frac{3}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.

30. State the Pythagorean identity solved for $\cos(\theta)$ in terms of $\sin(\theta)$.

    $\cos(\theta)=\pm\sqrt{1-\sin^2(\theta)}$. Rearrange Pythagorean identity; sign depends on quadrant.

31. Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\cos(\theta)=-\frac{4}{5}$.

    $\tan(\theta)=-\frac{3}{4}$. Direct division: $\tan=\frac{\sin}{\cos}=\frac{3/5}{-4/5}=-\frac{3}{4}$.

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

    $\cos(\theta)=-\frac{4}{5}$. Use $\cos^2=1-\sin^2=1-\frac{9}{25}=\frac{16}{25}$; QII means negative.

33. What is the sign of $\cos(\theta)$ in Quadrant III?

    $\cos(\theta)<0$. In QIII, $x$-values are negative on the unit circle.

34. What is the definition of $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$?

    $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Ratio of opposite to adjacent sides in a right triangle.

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

    $\sin(\theta)=\frac{3}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII sin positive.

36. Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\cos(\theta)=\frac{4}{5}$.

    $\tan(\theta)=\frac{3}{4}$. Direct division: $\tan=\frac{\sin}{\cos}=\frac{3/5}{4/5}=\frac{3}{4}$.

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

    $-\frac{5}{13}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{144}{169}=\frac{25}{169}$; negative in Q3.

38. What is $\sin^2(\theta)$ rewritten using $\cos(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?

    $\sin^2(\theta)=1-\cos^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\cos^2(\theta)$ from both sides.

39. State the Pythagorean identity solved for $\sin(\theta)$ in terms of $\cos(\theta)$.

    $\sin(\theta)=\pm\sqrt{1-\cos^2(\theta)}$. Rearrange Pythagorean identity; sign depends on quadrant.

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

    $-\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; negative in Q2.

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