Proving the Pythagorean Identity - Pre-Calculus
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What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant III?
What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant III?
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$\sin(\theta)<0,\ \cos(\theta)<0$. Both coordinates are negative in the third quadrant.
$\sin(\theta)<0,\ \cos(\theta)<0$. Both coordinates are negative in the third quadrant.
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What is the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$?
What is the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$?
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$\sin^2(\theta)+\cos^2(\theta)=1$. Fundamental identity derived from the unit circle equation $x^2+y^2=1$.
$\sin^2(\theta)+\cos^2(\theta)=1$. Fundamental identity derived from the unit circle equation $x^2+y^2=1$.
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What is $\cos^2(\theta)$ rewritten using $\sin(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?
What is $\cos^2(\theta)$ rewritten using $\sin(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?
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$\cos^2(\theta)=1-\sin^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\sin^2(\theta)$ from both sides.
$\cos^2(\theta)=1-\sin^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\sin^2(\theta)$ from both sides.
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What is $\sin^2(\theta)$ rewritten using $\cos(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?
What is $\sin^2(\theta)$ rewritten using $\cos(\theta)$ from $\sin^2(\theta)+\cos^2(\theta)=1$?
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$\sin^2(\theta)=1-\cos^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\cos^2(\theta)$ from both sides.
$\sin^2(\theta)=1-\cos^2(\theta)$. Rearrange the Pythagorean identity by subtracting $\cos^2(\theta)$ from both sides.
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What is the definition of $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$?
What is the definition of $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$?
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$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Ratio of opposite to adjacent sides in a right triangle.
$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Ratio of opposite to adjacent sides in a right triangle.
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What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant I?
What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant I?
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$\sin(\theta)>0,\ \cos(\theta)>0$. Both coordinates are positive in the first quadrant.
$\sin(\theta)>0,\ \cos(\theta)>0$. Both coordinates are positive in the first quadrant.
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What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant II?
What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant II?
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$\sin(\theta)>0,\ \cos(\theta)<0$. In Q2, $y$-values are positive and $x$-values are negative.
$\sin(\theta)>0,\ \cos(\theta)<0$. In Q2, $y$-values are positive and $x$-values are negative.
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What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant IV?
What are the signs of $\sin(\theta)$ and $\cos(\theta)$ in Quadrant IV?
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$\sin(\theta)<0,\ \cos(\theta)>0$. In Q4, $x$-values are positive and $y$-values are negative.
$\sin(\theta)<0,\ \cos(\theta)>0$. In Q4, $x$-values are positive and $y$-values are negative.
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What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I?
What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I?
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$\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; positive in Q1.
$\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; positive in Q1.
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What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II?
What is $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II?
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$-\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; negative in Q2.
$-\frac{4}{5}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{9}{25}=\frac{16}{25}$; negative in Q2.
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What is $\sin(\theta)$ if $\cos(\theta)=\frac{5}{13}$ and $\theta$ is in Quadrant I?
What is $\sin(\theta)$ if $\cos(\theta)=\frac{5}{13}$ and $\theta$ is in Quadrant I?
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$\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; positive in Q1.
$\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; positive in Q1.
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What is $\sin(\theta)$ if $\cos(\theta)=\frac{5}{13}$ and $\theta$ is in Quadrant IV?
What is $\sin(\theta)$ if $\cos(\theta)=\frac{5}{13}$ and $\theta$ is in Quadrant IV?
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$-\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; negative in Q4.
$-\frac{12}{13}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{25}{169}=\frac{144}{169}$; negative in Q4.
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What is $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I?
What is $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant I?
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$\frac{3}{4}$. Find $\cos(\theta)=\frac{4}{5}$ in Q1, then $\tan(\theta)=\frac{3/5}{4/5}=\frac{3}{4}$.
$\frac{3}{4}$. Find $\cos(\theta)=\frac{4}{5}$ in Q1, then $\tan(\theta)=\frac{3/5}{4/5}=\frac{3}{4}$.
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What is $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II?
What is $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II?
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$-\frac{3}{4}$. Find $\cos(\theta)=-\frac{4}{5}$ in Q2, then $\tan(\theta)=\frac{3/5}{-4/5}=-\frac{3}{4}$.
$-\frac{3}{4}$. Find $\cos(\theta)=-\frac{4}{5}$ in Q2, then $\tan(\theta)=\frac{3/5}{-4/5}=-\frac{3}{4}$.
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What is $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I?
What is $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I?
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$\frac{4}{5}$. From $\tan(\theta)=\frac{3}{4}$ and $\sin^2+\cos^2=1$ with Q1 signs.
$\frac{4}{5}$. From $\tan(\theta)=\frac{3}{4}$ and $\sin^2+\cos^2=1$ with Q1 signs.
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What is $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant III?
What is $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant III?
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$-\frac{3}{5}$. From $\tan(\theta)=\frac{3}{4}$ and Q3 signs, $\sin(\theta)=-\frac{3}{5}$.
$-\frac{3}{5}$. From $\tan(\theta)=\frac{3}{4}$ and Q3 signs, $\sin(\theta)=-\frac{3}{5}$.
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What is $\sin(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant II?
What is $\sin(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant II?
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$\frac{15}{17}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{64}{289}=\frac{225}{289}$; positive in Q2.
$\frac{15}{17}$. Use $\sin^2(\theta)=1-\cos^2(\theta)=1-\frac{64}{289}=\frac{225}{289}$; positive in Q2.
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What is $\tan(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant II?
What is $\tan(\theta)$ if $\cos(\theta)=-\frac{8}{17}$ and $\theta$ is in Quadrant II?
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$-\frac{15}{8}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{15/17}{-8/17}=-\frac{15}{8}$.
$-\frac{15}{8}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{15/17}{-8/17}=-\frac{15}{8}$.
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What is $\cos(\theta)$ if $\sin(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant III?
What is $\cos(\theta)$ if $\sin(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant III?
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$-\frac{5}{13}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{144}{169}=\frac{25}{169}$; negative in Q3.
$-\frac{5}{13}$. Use $\cos^2(\theta)=1-\sin^2(\theta)=1-\frac{144}{169}=\frac{25}{169}$; negative in Q3.
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What is $\tan(\theta)$ if $\sin(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant III?
What is $\tan(\theta)$ if $\sin(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant III?
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$\frac{12}{5}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{-12/13}{-5/13}=\frac{12}{5}$.
$\frac{12}{5}$. Calculate $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{-12/13}{-5/13}=\frac{12}{5}$.
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Find $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.
Find $\sin(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.
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$\sin(\theta)=\frac{3}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.
$\sin(\theta)=\frac{3}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.
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Find $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.
Find $\cos(\theta)$ if $\tan(\theta)=\frac{3}{4}$ and $\theta$ is in Quadrant I.
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$\cos(\theta)=\frac{4}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.
$\cos(\theta)=\frac{4}{5}$. From $\tan=\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve system; QI positive.
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Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\cos(\theta)=\frac{4}{5}$.
Find $\tan(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\cos(\theta)=\frac{4}{5}$.
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$\tan(\theta)=\frac{3}{4}$. Direct division: $\tan=\frac{\sin}{\cos}=\frac{3/5}{4/5}=\frac{3}{4}$.
$\tan(\theta)=\frac{3}{4}$. Direct division: $\tan=\frac{\sin}{\cos}=\frac{3/5}{4/5}=\frac{3}{4}$.
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Find $\sin(\theta)$ if $\cos(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant II.
Find $\sin(\theta)$ if $\cos(\theta)=-\frac{12}{13}$ and $\theta$ is in Quadrant II.
Tap to reveal answer
$\sin(\theta)=\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QII means positive.
$\sin(\theta)=\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QII means positive.
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Find $\sin(\theta)$ if $\cos(\theta)=\frac{12}{13}$ and $\theta$ is in Quadrant IV.
Find $\sin(\theta)$ if $\cos(\theta)=\frac{12}{13}$ and $\theta$ is in Quadrant IV.
Tap to reveal answer
$\sin(\theta)=-\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QIV means negative.
$\sin(\theta)=-\frac{5}{13}$. Use $\sin^2=1-\cos^2=1-\frac{144}{169}=\frac{25}{169}$; QIV means negative.
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Find $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II.
Find $\cos(\theta)$ if $\sin(\theta)=\frac{3}{5}$ and $\theta$ is in Quadrant II.
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$\cos(\theta)=-\frac{4}{5}$. Use $\cos^2=1-\sin^2=1-\frac{9}{25}=\frac{16}{25}$; QII means negative.
$\cos(\theta)=-\frac{4}{5}$. Use $\cos^2=1-\sin^2=1-\frac{9}{25}=\frac{16}{25}$; QII means negative.
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Find $\sin(\theta)$ if $\tan(\theta)=-\frac{3}{4}$ and $\theta$ is in Quadrant II.
Find $\sin(\theta)$ if $\tan(\theta)=-\frac{3}{4}$ and $\theta$ is in Quadrant II.
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$\sin(\theta)=\frac{3}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII sin positive.
$\sin(\theta)=\frac{3}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII sin positive.
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Find $\cos(\theta)$ if $\tan(\theta)=-\frac{3}{4}$ and $\theta$ is in Quadrant II.
Find $\cos(\theta)$ if $\tan(\theta)=-\frac{3}{4}$ and $\theta$ is in Quadrant II.
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$\cos(\theta)=-\frac{4}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII cos negative.
$\cos(\theta)=-\frac{4}{5}$. From $\tan=-\frac{3}{4}$ and $\sin^2+\cos^2=1$, solve; QII cos negative.
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What is the sign of $\tan(\theta)$ in Quadrant III?
What is the sign of $\tan(\theta)$ in Quadrant III?
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$\tan(\theta)>0$. Since both $\sin<0$ and $\cos<0$ in QIII, their ratio is positive.
$\tan(\theta)>0$. Since both $\sin<0$ and $\cos<0$ in QIII, their ratio is positive.
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What is the sign of $\cos(\theta)$ in Quadrant III?
What is the sign of $\cos(\theta)$ in Quadrant III?
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$\cos(\theta)<0$. In QIII, $x$-values are negative on the unit circle.
$\cos(\theta)<0$. In QIII, $x$-values are negative on the unit circle.
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