Sine and Cosine of Complementary Angles - Pre-Calculus
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What is the complementary-angle identity for tangent in a right triangle?
What is the complementary-angle identity for tangent in a right triangle?
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$\tan(\theta)=\cot(90^\circ-\theta)$. Complementary angles have reciprocal tan/cot values in right triangles.
$\tan(\theta)=\cot(90^\circ-\theta)$. Complementary angles have reciprocal tan/cot values in right triangles.
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What does AA similarity mean for triangles (state the criterion)?
What does AA similarity mean for triangles (state the criterion)?
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If two angles match, the triangles are similar. AA (Angle-Angle) similarity requires only two pairs of equal angles.
If two angles match, the triangles are similar. AA (Angle-Angle) similarity requires only two pairs of equal angles.
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What is the definition of $ an( heta)$ in a right triangle for acute $ heta$?
What is the definition of $ an( heta)$ in a right triangle for acute $ heta$?
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$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of the opposite side to the adjacent side.
$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of the opposite side to the adjacent side.
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What is the definition of similar triangles in terms of angles and side ratios?
What is the definition of similar triangles in terms of angles and side ratios?
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Corresponding angles equal and corresponding sides proportional. Similar triangles have matching angles and proportional corresponding sides.
Corresponding angles equal and corresponding sides proportional. Similar triangles have matching angles and proportional corresponding sides.
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What is the definition of $ rac{\text{opposite}}{\text{hypotenuse}}$ for acute angle $ heta$?
What is the definition of $ rac{\text{opposite}}{\text{hypotenuse}}$ for acute angle $ heta$?
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$\sin(\theta)$. Sine is defined as opposite over hypotenuse in a right triangle.
$\sin(\theta)$. Sine is defined as opposite over hypotenuse in a right triangle.
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What is the definition of $ rac{\text{adjacent}}{\text{hypotenuse}}$ for acute angle $ heta$?
What is the definition of $ rac{\text{adjacent}}{\text{hypotenuse}}$ for acute angle $ heta$?
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$\cos(\theta)$. Cosine is defined as adjacent over hypotenuse in a right triangle.
$\cos(\theta)$. Cosine is defined as adjacent over hypotenuse in a right triangle.
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What is the definition of $ rac{\text{hypotenuse}}{\text{opposite}}$ for acute angle $ heta$?
What is the definition of $ rac{\text{hypotenuse}}{\text{opposite}}$ for acute angle $ heta$?
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$\csc(\theta)$. Cosecant is the reciprocal of sine: hypotenuse over opposite.
$\csc(\theta)$. Cosecant is the reciprocal of sine: hypotenuse over opposite.
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What is the definition of $ rac{\text{adjacent}}{\text{opposite}}$ for acute angle $ heta$?
What is the definition of $ rac{\text{adjacent}}{\text{opposite}}$ for acute angle $ heta$?
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$\cot(\theta)$. Cotangent is the reciprocal of tangent: adjacent over opposite.
$\cot(\theta)$. Cotangent is the reciprocal of tangent: adjacent over opposite.
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What relationship between right triangles justifies trig ratios depending only on angle $ heta$?
What relationship between right triangles justifies trig ratios depending only on angle $ heta$?
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Right triangles with the same acute angle are similar. AA similarity ensures all right triangles with same acute angle have equal trig ratios.
Right triangles with the same acute angle are similar. AA similarity ensures all right triangles with same acute angle have equal trig ratios.
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What is the reciprocal identity for sine and cosecant?
What is the reciprocal identity for sine and cosecant?
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$\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant and sine are reciprocals of each other.
$\csc(\theta)=\frac{1}{\sin(\theta)}$. Cosecant and sine are reciprocals of each other.
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Identify the value of $\sin(\theta)$ if opposite $=3$ and hypotenuse $=5$ for acute $\theta$.
Identify the value of $\sin(\theta)$ if opposite $=3$ and hypotenuse $=5$ for acute $\theta$.
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$\sin(\theta)=\frac{3}{5}$. Apply sine definition: opposite over hypotenuse.
$\sin(\theta)=\frac{3}{5}$. Apply sine definition: opposite over hypotenuse.
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What is $\cos(\theta)$ if $\sin(\theta)=\frac{5}{13}$ and $\theta$ is acute?
What is $\cos(\theta)$ if $\sin(\theta)=\frac{5}{13}$ and $\theta$ is acute?
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$\cos(\theta)=\frac{12}{13}$. Use $\sin^2(\theta)+\cos^2(\theta)=1$ with $\sin(\theta)=\frac{5}{13}$.
$\cos(\theta)=\frac{12}{13}$. Use $\sin^2(\theta)+\cos^2(\theta)=1$ with $\sin(\theta)=\frac{5}{13}$.
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Identify the value of $\tan(\theta)$ if opposite $=7$ and adjacent $=4$ for acute $\theta$.
Identify the value of $\tan(\theta)$ if opposite $=7$ and adjacent $=4$ for acute $\theta$.
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$\tan(\theta)=\frac{7}{4}$. Apply tangent definition: opposite over adjacent.
$\tan(\theta)=\frac{7}{4}$. Apply tangent definition: opposite over adjacent.
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Identify the value of $\cos(\theta)$ if adjacent $=12$ and hypotenuse $=13$ for acute $\theta$.
Identify the value of $\cos(\theta)$ if adjacent $=12$ and hypotenuse $=13$ for acute $\theta$.
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$\cos(\theta)=\frac{12}{13}$. Apply cosine definition: adjacent over hypotenuse.
$\cos(\theta)=\frac{12}{13}$. Apply cosine definition: adjacent over hypotenuse.
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What is the reciprocal identity for cosine and secant?
What is the reciprocal identity for cosine and secant?
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$\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant and cosine are reciprocals of each other.
$\sec(\theta)=\frac{1}{\cos(\theta)}$. Secant and cosine are reciprocals of each other.
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What is the reciprocal identity for tangent and cotangent?
What is the reciprocal identity for tangent and cotangent?
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$\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent and tangent are reciprocals of each other.
$\cot(\theta)=\frac{1}{\tan(\theta)}$. Cotangent and tangent are reciprocals of each other.
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What is the complementary-angle identity for sine in a right triangle?
What is the complementary-angle identity for sine in a right triangle?
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$\sin(\theta)=\cos(90^\circ-\theta)$. In a right triangle, complementary angles swap opposite and adjacent sides.
$\sin(\theta)=\cos(90^\circ-\theta)$. In a right triangle, complementary angles swap opposite and adjacent sides.
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What is the definition of $tan( heta)$ in a right triangle for acute $ heta$?
What is the definition of $tan( heta)$ in a right triangle for acute $ heta$?
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$tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of opposite to adjacent sides.
$tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. Tangent is the ratio of opposite to adjacent sides.
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Identify the side used as the hypotenuse in a right triangle.
Identify the side used as the hypotenuse in a right triangle.
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The side opposite the $90^\circ$ angle (the longest side). The hypotenuse is always opposite the right angle.
The side opposite the $90^\circ$ angle (the longest side). The hypotenuse is always opposite the right angle.
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Identify the side called opposite relative to an acute angle $ heta$ in a right triangle.
Identify the side called opposite relative to an acute angle $ heta$ in a right triangle.
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The side across from $\theta$. The opposite side doesn't touch the angle $\theta$.
The side across from $\theta$. The opposite side doesn't touch the angle $\theta$.
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Identify the side called adjacent relative to an acute angle $ heta$ in a right triangle.
Identify the side called adjacent relative to an acute angle $ heta$ in a right triangle.
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The non-hypotenuse side that touches $\theta$. Adjacent means next to the angle, excluding the hypotenuse.
The non-hypotenuse side that touches $\theta$. Adjacent means next to the angle, excluding the hypotenuse.
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Find $sin(\theta)$ if opposite $=9$ and hypotenuse $=15$.
Find $sin(\theta)$ if opposite $=9$ and hypotenuse $=15$.
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$sin(\theta)=\frac{9}{15}=\frac{3}{5}$. Apply the sine ratio formula and simplify the fraction.
$sin(\theta)=\frac{9}{15}=\frac{3}{5}$. Apply the sine ratio formula and simplify the fraction.
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Find $cos(\theta)$ if adjacent $=12$ and hypotenuse $=13$.
Find $cos(\theta)$ if adjacent $=12$ and hypotenuse $=13$.
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$cos(\theta)=\frac{12}{13}$. Apply the cosine ratio formula directly.
$cos(\theta)=\frac{12}{13}$. Apply the cosine ratio formula directly.
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What is the definition of $cot( heta)$ in a right triangle for acute $ heta$?
What is the definition of $cot( heta)$ in a right triangle for acute $ heta$?
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$cot(\theta)=\frac{\text{adjacent}}{\text{opposite}}$. Cotangent is the reciprocal of tangent.
$cot(\theta)=\frac{\text{adjacent}}{\text{opposite}}$. Cotangent is the reciprocal of tangent.
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What is the definition of $sec( heta)$ in a right triangle for acute $ heta$?
What is the definition of $sec( heta)$ in a right triangle for acute $ heta$?
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$sec(\theta)=\frac{\text{hypotenuse}}{\text{adjacent}}$. Secant is the reciprocal of cosine.
$sec(\theta)=\frac{\text{hypotenuse}}{\text{adjacent}}$. Secant is the reciprocal of cosine.
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What is the definition of $csc( heta)$ in a right triangle for acute $ heta$?
What is the definition of $csc( heta)$ in a right triangle for acute $ heta$?
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$csc(\theta)=\frac{\text{hypotenuse}}{\text{opposite}}$. Cosecant is the reciprocal of sine.
$csc(\theta)=\frac{\text{hypotenuse}}{\text{opposite}}$. Cosecant is the reciprocal of sine.
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What is the complementary-angle identity for cosine and sine in a right triangle?
What is the complementary-angle identity for cosine and sine in a right triangle?
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$cos(\theta)=sin(90^\circ-\theta)$. Complementary angles in right triangles interchange sine and cosine.
$cos(\theta)=sin(90^\circ-\theta)$. Complementary angles in right triangles interchange sine and cosine.
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What is the definition of $cos( heta)$ in a right triangle for acute $ heta$?
What is the definition of $cos( heta)$ in a right triangle for acute $ heta$?
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$cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Cosine is the ratio of the adjacent side to the hypotenuse.
$cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. Cosine is the ratio of the adjacent side to the hypotenuse.
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What is the definition of $sin( heta)$ in a right triangle for acute $ heta$?
What is the definition of $sin( heta)$ in a right triangle for acute $ heta$?
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$sin( heta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Sine is the ratio of the opposite side to the hypotenuse.
$sin( heta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Sine is the ratio of the opposite side to the hypotenuse.
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What does similarity imply about ratios of corresponding sides in similar triangles?
What does similarity imply about ratios of corresponding sides in similar triangles?
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Corresponding side ratios are equal (a constant scale factor). Similar triangles have proportional sides with the same ratio.
Corresponding side ratios are equal (a constant scale factor). Similar triangles have proportional sides with the same ratio.
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