Sine and Cosine of Complementary Angles - Geometry
Card 1 of 30
What is $ ext{tan}( heta)$ if the opposite side is $7$ and the adjacent side is $2$?
What is $ ext{tan}( heta)$ if the opposite side is $7$ and the adjacent side is $2$?
Tap to reveal answer
$\text{tan}(\theta)=\frac{7}{2}$. Apply tangent definition with given values.
$\text{tan}(\theta)=\frac{7}{2}$. Apply tangent definition with given values.
← Didn't Know|Knew It →
What is $ ext{csc}( heta)$ if $ ext{sin}( heta)=\frac{5}{13}$?
What is $ ext{csc}( heta)$ if $ ext{sin}( heta)=\frac{5}{13}$?
Tap to reveal answer
$\text{csc}(\theta)=\frac{13}{5}$. Take reciprocal of given sine value.
$\text{csc}(\theta)=\frac{13}{5}$. Take reciprocal of given sine value.
← Didn't Know|Knew It →
What is $ ext{cot}( heta)$ if $ ext{tan}( heta)=\frac{9}{4}$?
What is $ ext{cot}( heta)$ if $ ext{tan}( heta)=\frac{9}{4}$?
Tap to reveal answer
$\text{cot}(\theta)=\frac{4}{9}$. Take reciprocal of given tangent value.
$\text{cot}(\theta)=\frac{4}{9}$. Take reciprocal of given tangent value.
← Didn't Know|Knew It →
What is $ ext{tan}( heta)$ if $ ext{sin}( heta)=\frac{3}{5}$ and $ ext{cos}( heta)=\frac{4}{5}$?
What is $ ext{tan}( heta)$ if $ ext{sin}( heta)=\frac{3}{5}$ and $ ext{cos}( heta)=\frac{4}{5}$?
Tap to reveal answer
$\text{tan}(\theta)=\frac{3}{4}$. Use quotient identity $\text{tan}(\theta)=\frac{\text{sin}(\theta)}{\text{cos}(\theta)}$.
$\text{tan}(\theta)=\frac{3}{4}$. Use quotient identity $\text{tan}(\theta)=\frac{\text{sin}(\theta)}{\text{cos}(\theta)}$.
← Didn't Know|Knew It →
What is $ ext{sin}(90^{\circ}-\theta)$ in terms of $ ext{cos}(\theta)$?
What is $ ext{sin}(90^{\circ}-\theta)$ in terms of $ ext{cos}(\theta)$?
Tap to reveal answer
$\text{sin}(90^{\circ}-\theta)=\text{cos}(\theta)$. Sine of complement equals cosine of original angle.
$\text{sin}(90^{\circ}-\theta)=\text{cos}(\theta)$. Sine of complement equals cosine of original angle.
← Didn't Know|Knew It →
What is $ ext{cos}(90^{\circ}-\theta)$ in terms of $ ext{sin}(\theta)$?
What is $ ext{cos}(90^{\circ}-\theta)$ in terms of $ ext{sin}(\theta)$?
Tap to reveal answer
$\text{cos}(90^{\circ}-\theta)=\text{sin}(\theta)$. Cosine of complement equals sine of original angle.
$\text{cos}(90^{\circ}-\theta)=\text{sin}(\theta)$. Cosine of complement equals sine of original angle.
← Didn't Know|Knew It →
What is $ ext{tan}(90^{\circ}-\theta)$ in terms of $ ext{cot}(\theta)$?
What is $ ext{tan}(90^{\circ}-\theta)$ in terms of $ ext{cot}(\theta)$?
Tap to reveal answer
$\text{tan}(90^{\circ}-\theta)=\text{cot}(\theta)$. Tangent of complement equals cotangent of original angle.
$\text{tan}(90^{\circ}-\theta)=\text{cot}(\theta)$. Tangent of complement equals cotangent of original angle.
← Didn't Know|Knew It →
What is $ ext{cot}(90^{\circ}-\theta)$ in terms of $ ext{tan}(\theta)$?
What is $ ext{cot}(90^{\circ}-\theta)$ in terms of $ ext{tan}(\theta)$?
Tap to reveal answer
$\text{cot}(90^{\circ}-\theta)=\text{tan}(\theta)$. Cotangent of complement equals tangent of original angle.
$\text{cot}(90^{\circ}-\theta)=\text{tan}(\theta)$. Cotangent of complement equals tangent of original angle.
← Didn't Know|Knew It →
What is $ ext{sec}(90^{\circ}-\theta)$ in terms of $ ext{csc}(\theta)$?
What is $ ext{sec}(90^{\circ}-\theta)$ in terms of $ ext{csc}(\theta)$?
Tap to reveal answer
$\text{sec}(90^{\circ}-\theta)=\text{csc}(\theta)$. Secant of complement equals cosecant of original angle.
$\text{sec}(90^{\circ}-\theta)=\text{csc}(\theta)$. Secant of complement equals cosecant of original angle.
← Didn't Know|Knew It →
What is $ ext{sec}( heta)$ if $ ext{cos}( heta)=\frac{11}{15}$?
What is $ ext{sec}( heta)$ if $ ext{cos}( heta)=\frac{11}{15}$?
Tap to reveal answer
$\text{sec}(\theta)=\frac{15}{11}$. Take reciprocal of given cosine value.
$\text{sec}(\theta)=\frac{15}{11}$. Take reciprocal of given cosine value.
← Didn't Know|Knew It →
What is $ ext{csc}(90^{\circ}-\theta)$ in terms of $ ext{sec}(\theta)$?
What is $ ext{csc}(90^{\circ}-\theta)$ in terms of $ ext{sec}(\theta)$?
Tap to reveal answer
$\text{csc}(90^{\circ}-\theta)=\text{sec}(\theta)$. Cosecant of complement equals secant of original angle.
$\text{csc}(90^{\circ}-\theta)=\text{sec}(\theta)$. Cosecant of complement equals secant of original angle.
← Didn't Know|Knew It →
What is $ ext{cos}(\theta)$ if $ ext{sin}(\theta)=\frac{3}{5}$ and $\theta$ is acute?
What is $ ext{cos}(\theta)$ if $ ext{sin}(\theta)=\frac{3}{5}$ and $\theta$ is acute?
Tap to reveal answer
$\text{cos}(\theta)=\frac{4}{5}$. Use Pythagorean identity: $\text{cos}^2(\theta)=1-\text{sin}^2(\theta)=\frac{16}{25}$.
$\text{cos}(\theta)=\frac{4}{5}$. Use Pythagorean identity: $\text{cos}^2(\theta)=1-\text{sin}^2(\theta)=\frac{16}{25}$.
← Didn't Know|Knew It →
What is $ ext{sin}(\theta)$ if $ ext{cos}(\theta)=\frac{12}{13}$ and $\theta$ is acute?
What is $ ext{sin}(\theta)$ if $ ext{cos}(\theta)=\frac{12}{13}$ and $\theta$ is acute?
Tap to reveal answer
$\text{sin}(\theta)=\frac{5}{13}$. Use Pythagorean identity: $\text{sin}^2(\theta)=1-\text{cos}^2(\theta)=\frac{25}{169}$.
$\text{sin}(\theta)=\frac{5}{13}$. Use Pythagorean identity: $\text{sin}^2(\theta)=1-\text{cos}^2(\theta)=\frac{25}{169}$.
← Didn't Know|Knew It →
What is $ ext{sin}(\theta)$ if $ ext{cos}(\theta)=\frac{7}{25}$ and $\theta$ is acute?
What is $ ext{sin}(\theta)$ if $ ext{cos}(\theta)=\frac{7}{25}$ and $\theta$ is acute?
Tap to reveal answer
$\text{sin}(\theta)=\frac{24}{25}$. Use Pythagorean identity: $\text{sin}^2(\theta)=1-\text{cos}^2(\theta)=\frac{576}{625}$.
$\text{sin}(\theta)=\frac{24}{25}$. Use Pythagorean identity: $\text{sin}^2(\theta)=1-\text{cos}^2(\theta)=\frac{576}{625}$.
← Didn't Know|Knew It →
What is $ ext{cos}(\theta)$ if $ ext{sin}(\theta)=\frac{8}{17}$ and $\theta$ is acute?
What is $ ext{cos}(\theta)$ if $ ext{sin}(\theta)=\frac{8}{17}$ and $\theta$ is acute?
Tap to reveal answer
$\text{cos}(\theta)=\frac{15}{17}$. Use Pythagorean identity: $\text{cos}^2(\theta)=1-\text{sin}^2(\theta)=\frac{225}{289}$.
$\text{cos}(\theta)=\frac{15}{17}$. Use Pythagorean identity: $\text{cos}^2(\theta)=1-\text{sin}^2(\theta)=\frac{225}{289}$.
← Didn't Know|Knew It →
What is $ ext{tan}(\theta)$ if $ ext{sin}(\theta)=\frac{5}{13}$ and $ ext{cos}(\theta)=\frac{12}{13}$?
What is $ ext{tan}(\theta)$ if $ ext{sin}(\theta)=\frac{5}{13}$ and $ ext{cos}(\theta)=\frac{12}{13}$?
Tap to reveal answer
$\text{tan}(\theta)=\frac{5}{12}$. Use quotient identity: $\text{tan}(\theta)=\frac{\text{sin}(\theta)}{\text{cos}(\theta)}$.
$\text{tan}(\theta)=\frac{5}{12}$. Use quotient identity: $\text{tan}(\theta)=\frac{\text{sin}(\theta)}{\text{cos}(\theta)}$.
← Didn't Know|Knew It →
What is $ ext{cos}(\theta)$ if the sides relative to $ heta$ are adjacent $=9$ and hypotenuse $=15$?
What is $ ext{cos}(\theta)$ if the sides relative to $ heta$ are adjacent $=9$ and hypotenuse $=15$?
Tap to reveal answer
$\text{cos}(\theta)=\frac{3}{5}$. Simplify fraction: $\frac{9}{15}=\frac{3}{5}$.
$\text{cos}(\theta)=\frac{3}{5}$. Simplify fraction: $\frac{9}{15}=\frac{3}{5}$.
← Didn't Know|Knew It →
What is $ ext{tan}(\theta)$ if the sides relative to $ heta$ are opposite $=12$ and adjacent $=5$?
What is $ ext{tan}(\theta)$ if the sides relative to $ heta$ are opposite $=12$ and adjacent $=5$?
Tap to reveal answer
$\text{tan}(\theta)=\frac{12}{5}$. Apply tangent definition with given side lengths.
$\text{tan}(\theta)=\frac{12}{5}$. Apply tangent definition with given side lengths.
← Didn't Know|Knew It →
What is $ ext{sin}(\theta)$ if a similar right triangle scales all sides by $k$?
What is $ ext{sin}(\theta)$ if a similar right triangle scales all sides by $k$?
Tap to reveal answer
$\text{sin}(\theta)$ is unchanged (ratio stays constant). Scaling preserves all trigonometric ratios.
$\text{sin}(\theta)$ is unchanged (ratio stays constant). Scaling preserves all trigonometric ratios.
← Didn't Know|Knew It →
What is the value of $ rac{\text{sin}(\theta)}{\text{cos}(\theta)}$ in terms of $ ext{tan}(\theta)$?
What is the value of $ rac{\text{sin}(\theta)}{\text{cos}(\theta)}$ in terms of $ ext{tan}(\theta)$?
Tap to reveal answer
$\frac{\text{sin}(\theta)}{\text{cos}(\theta)}=\text{tan}(\theta)$. This is the definition of tangent function.
$\frac{\text{sin}(\theta)}{\text{cos}(\theta)}=\text{tan}(\theta)$. This is the definition of tangent function.
← Didn't Know|Knew It →
Identify the error: $\text{sin}(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.
Identify the error: $\text{sin}(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$.
Tap to reveal answer
Correct: $\text{sin}(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Sine uses opposite over hypotenuse, not adjacent.
Correct: $\text{sin}(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Sine uses opposite over hypotenuse, not adjacent.
← Didn't Know|Knew It →
What quotient identity expresses $\tan(\theta)$ using $\sin(\theta)$ and $\cos(\theta)$?
What quotient identity expresses $\tan(\theta)$ using $\sin(\theta)$ and $\cos(\theta)$?
Tap to reveal answer
$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Tangent equals sine divided by cosine.
$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$. Tangent equals sine divided by cosine.
← Didn't Know|Knew It →
What is the definition of $\frac{\text{hypotenuse}}{\text{adjacent}}$ for an acute angle $\theta$?
What is the definition of $\frac{\text{hypotenuse}}{\text{adjacent}}$ for an acute angle $\theta$?
Tap to reveal answer
$\frac{\text{hypotenuse}}{\text{adjacent}}=\sec(\theta)$. Reciprocal of cosine function definition.
$\frac{\text{hypotenuse}}{\text{adjacent}}=\sec(\theta)$. Reciprocal of cosine function definition.
← Didn't Know|Knew It →
What is $\text{sin}(\theta)$ if the sides relative to $\theta$ are opposite =6 and hypotenuse =10?
What is $\text{sin}(\theta)$ if the sides relative to $\theta$ are opposite =6 and hypotenuse =10?
Tap to reveal answer
$\text{sin}(\theta)=\frac{3}{5}$. Simplify fraction: $\frac{6}{10}=\frac{3}{5}$.
$\text{sin}(\theta)=\frac{3}{5}$. Simplify fraction: $\frac{6}{10}=\frac{3}{5}$.
← Didn't Know|Knew It →
What theorem justifies that any two right triangles with the same acute angle are similar?
What theorem justifies that any two right triangles with the same acute angle are similar?
Tap to reveal answer
AA similarity (both have a right angle and one equal acute angle). Two triangles with two equal angles must be similar.
AA similarity (both have a right angle and one equal acute angle). Two triangles with two equal angles must be similar.
← Didn't Know|Knew It →
What does similarity imply about ratios of corresponding side lengths in similar triangles?
What does similarity imply about ratios of corresponding side lengths in similar triangles?
Tap to reveal answer
Corresponding side ratios are equal (a constant scale factor). Similar triangles have proportional sides with constant ratio.
Corresponding side ratios are equal (a constant scale factor). Similar triangles have proportional sides with constant ratio.
← Didn't Know|Knew It →
What is the definition of $ rac{ ext{opposite}}{ ext{hypotenuse}}$ for an acute angle $ heta$?
What is the definition of $ rac{ ext{opposite}}{ ext{hypotenuse}}$ for an acute angle $ heta$?
Tap to reveal answer
$ rac{ ext{opposite}}{ ext{hypotenuse}}= ext{sin}( heta)$. Basic trigonometric ratio using opposite side and hypotenuse.
$ rac{ ext{opposite}}{ ext{hypotenuse}}= ext{sin}( heta)$. Basic trigonometric ratio using opposite side and hypotenuse.
← Didn't Know|Knew It →
What is the definition of $ rac{ ext{adjacent}}{ ext{hypotenuse}}$ for an acute angle $ heta$?
What is the definition of $ rac{ ext{adjacent}}{ ext{hypotenuse}}$ for an acute angle $ heta$?
Tap to reveal answer
$ rac{ ext{adjacent}}{ ext{hypotenuse}}= ext{cos}( heta)$. Basic trigonometric ratio using adjacent side and hypotenuse.
$ rac{ ext{adjacent}}{ ext{hypotenuse}}= ext{cos}( heta)$. Basic trigonometric ratio using adjacent side and hypotenuse.
← Didn't Know|Knew It →
What is the definition of $ rac{ ext{adjacent}}{ ext{opposite}}$ for an acute angle $ heta$?
What is the definition of $ rac{ ext{adjacent}}{ ext{opposite}}$ for an acute angle $ heta$?
Tap to reveal answer
$ rac{ ext{adjacent}}{ ext{opposite}}= ext{cot}( heta)$. Reciprocal of tangent function definition.
$ rac{ ext{adjacent}}{ ext{opposite}}= ext{cot}( heta)$. Reciprocal of tangent function definition.
← Didn't Know|Knew It →
What is the reciprocal identity relating $ ext{sin}( heta)$ and $ ext{csc}( heta)$?
What is the reciprocal identity relating $ ext{sin}( heta)$ and $ ext{csc}( heta)$?
Tap to reveal answer
$ ext{csc}( heta)= rac{1}{ ext{sin}( heta)}$. Cosecant and sine are reciprocal functions.
$ ext{csc}( heta)= rac{1}{ ext{sin}( heta)}$. Cosecant and sine are reciprocal functions.
← Didn't Know|Knew It →