Proving, Applying Laws of Sines/Cosines - Geometry
Card 1 of 30
Identify the height to side $a$ in terms of side $b$ and included angle $C$.
Identify the height to side $a$ in terms of side $b$ and included angle $C$.
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$h=b\sin(C)$. From right triangle with hypotenuse $b$ and angle $C$.
$h=b\sin(C)$. From right triangle with hypotenuse $b$ and angle $C$.
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What right-triangle trig definition gives height when hypotenuse is $a$ and angle is $C$?
What right-triangle trig definition gives height when hypotenuse is $a$ and angle is $C$?
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$\sin(C)=\frac{h}{a}$. Sine equals opposite over hypotenuse in right triangles.
$\sin(C)=\frac{h}{a}$. Sine equals opposite over hypotenuse in right triangles.
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What does $h$ represent in the formula $A=\frac{1}{2}bh$?
What does $h$ represent in the formula $A=\frac{1}{2}bh$?
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$h$ is the perpendicular height to base $b$. The perpendicular distance from vertex to base.
$h$ is the perpendicular height to base $b$. The perpendicular distance from vertex to base.
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What is the standard base-height area formula for any triangle?
What is the standard base-height area formula for any triangle?
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$A=\frac{1}{2}bh$. Uses base times height, divided by 2.
$A=\frac{1}{2}bh$. Uses base times height, divided by 2.
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What auxiliary line is drawn to derive $A=\frac{1}{2}ab\sin(C)$?
What auxiliary line is drawn to derive $A=\frac{1}{2}ab\sin(C)$?
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A perpendicular from a vertex to the opposite side (or its extension). Creates a right triangle to find height using trigonometry.
A perpendicular from a vertex to the opposite side (or its extension). Creates a right triangle to find height using trigonometry.
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What is the area formula using two sides $a,b$ and included angle $C$?
What is the area formula using two sides $a,b$ and included angle $C$?
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$A=\frac{1}{2}ab\sin(C)$. Uses two sides and the included angle between them.
$A=\frac{1}{2}ab\sin(C)$. Uses two sides and the included angle between them.
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In $A=\frac{1}{2}ab\sin(C)$, what does $C$ represent?
In $A=\frac{1}{2}ab\sin(C)$, what does $C$ represent?
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The included angle between sides $a$ and $b$. The angle between the two given sides $a$ and $b$.
The included angle between sides $a$ and $b$. The angle between the two given sides $a$ and $b$.
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Identify the height to side $a$ in terms of side $b$ and included angle $C$.
Identify the height to side $a$ in terms of side $b$ and included angle $C$.
Tap to reveal answer
$h=b\sin(C)$. From right triangle with hypotenuse $b$ and angle $C$.
$h=b\sin(C)$. From right triangle with hypotenuse $b$ and angle $C$.
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Identify the height to side $b$ in terms of side $a$ and included angle $C$.
Identify the height to side $b$ in terms of side $a$ and included angle $C$.
Tap to reveal answer
$h=a\sin(C)$. From right triangle with hypotenuse $a$ and angle $C$.
$h=a\sin(C)$. From right triangle with hypotenuse $a$ and angle $C$.
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What right-triangle trig definition gives height when hypotenuse is $b$ and angle is $C$?
What right-triangle trig definition gives height when hypotenuse is $b$ and angle is $C$?
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$\sin(C)=\frac{h}{b}$. Sine equals opposite over hypotenuse in right triangles.
$\sin(C)=\frac{h}{b}$. Sine equals opposite over hypotenuse in right triangles.
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What right-triangle trig definition gives height when hypotenuse is $a$ and angle is $C$?
What right-triangle trig definition gives height when hypotenuse is $a$ and angle is $C$?
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$\sin(C)=\frac{h}{a}$. Sine equals opposite over hypotenuse in right triangles.
$\sin(C)=\frac{h}{a}$. Sine equals opposite over hypotenuse in right triangles.
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What is $h$ after solving $\sin(C)=\frac{h}{b}$ for $h$?
What is $h$ after solving $\sin(C)=\frac{h}{b}$ for $h$?
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$h=b\sin(C)$. Multiply both sides by $b$ to isolate $h$.
$h=b\sin(C)$. Multiply both sides by $b$ to isolate $h$.
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What is $h$ after solving $\sin(C)=\frac{h}{a}$ for $h$?
What is $h$ after solving $\sin(C)=\frac{h}{a}$ for $h$?
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$h=a\sin(C)$. Multiply both sides by $a$ to isolate $h$.
$h=a\sin(C)$. Multiply both sides by $a$ to isolate $h$.
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What area expression results from substituting $h=b\sin(C)$ into $A=\frac{1}{2}ah$?
What area expression results from substituting $h=b\sin(C)$ into $A=\frac{1}{2}ah$?
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$A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.
$A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.
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What area expression results from substituting $h=a\sin(C)$ into $A=\frac{1}{2}bh$?
What area expression results from substituting $h=a\sin(C)$ into $A=\frac{1}{2}bh$?
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$A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.
$A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.
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Which trigonometric function appears in the two-sides-included-angle area formula?
Which trigonometric function appears in the two-sides-included-angle area formula?
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$\sin$. Sine relates the height to the sides and included angle.
$\sin$. Sine relates the height to the sides and included angle.
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What is $\sin(90^\circ)$, and what does it imply for area when $C=90^\circ$?
What is $\sin(90^\circ)$, and what does it imply for area when $C=90^\circ$?
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$\sin(90^\circ)=1$, so $A=\frac{1}{2}ab$. Right angle makes formula reduce to standard rectangle area.
$\sin(90^\circ)=1$, so $A=\frac{1}{2}ab$. Right angle makes formula reduce to standard rectangle area.
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What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=0^\circ$?
What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=0^\circ$?
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$A=0$. Sine of zero equals zero, making area zero (degenerate).
$A=0$. Sine of zero equals zero, making area zero (degenerate).
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What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=180^\circ$?
What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=180^\circ$?
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$A=0$. Sine of 180° equals zero, making area zero (degenerate).
$A=0$. Sine of 180° equals zero, making area zero (degenerate).
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What is the key right-triangle relationship between opposite leg, hypotenuse, and sine?
What is the key right-triangle relationship between opposite leg, hypotenuse, and sine?
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$\text{opposite}=\text{hypotenuse}\cdot\sin(\text{angle})$. Basic trigonometric identity for right triangles.
$\text{opposite}=\text{hypotenuse}\cdot\sin(\text{angle})$. Basic trigonometric identity for right triangles.
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What construction handles the case when the altitude from a vertex falls outside the triangle?
What construction handles the case when the altitude from a vertex falls outside the triangle?
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Extend the opposite side and drop a perpendicular to the extension. Handles obtuse triangles where altitude falls outside.
Extend the opposite side and drop a perpendicular to the extension. Handles obtuse triangles where altitude falls outside.
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Which identity justifies using $\sin(C)$ even if the altitude meets an extension and forms angle $180^\circ-C$?
Which identity justifies using $\sin(C)$ even if the altitude meets an extension and forms angle $180^\circ-C$?
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$\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity for sine function.
$\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity for sine function.
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What is $\sin(180^\circ-C)$ in terms of $\sin(C)$?
What is $\sin(180^\circ-C)$ in terms of $\sin(C)$?
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$\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity: sine values are equal.
$\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity: sine values are equal.
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Find $A$ if $a=10$, $b=6$, and $C=30^\circ$ using $A=\frac{1}{2}ab\sin(C)$.
Find $A$ if $a=10$, $b=6$, and $C=30^\circ$ using $A=\frac{1}{2}ab\sin(C)$.
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$A=15$. $\frac{1}{2}(10)(6)\sin(30°) = \frac{1}{2}(10)(6)(\frac{1}{2}) = 15$
$A=15$. $\frac{1}{2}(10)(6)\sin(30°) = \frac{1}{2}(10)(6)(\frac{1}{2}) = 15$
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Find $A$ if $a=8$, $b=5$, and $C=90^\circ$ using $A=\frac{1}{2}ab\sin(C)$.
Find $A$ if $a=8$, $b=5$, and $C=90^\circ$ using $A=\frac{1}{2}ab\sin(C)$.
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$A=20$. $\frac{1}{2}(8)(5)\sin(90°) = \frac{1}{2}(8)(5)(1) = 20$
$A=20$. $\frac{1}{2}(8)(5)\sin(90°) = \frac{1}{2}(8)(5)(1) = 20$
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Find $A$ if $a=12$, $b=7$, and $C=60^\circ$ with $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.
Find $A$ if $a=12$, $b=7$, and $C=60^\circ$ with $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.
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$A=21\sqrt{3}$. $\frac{1}{2}(12)(7)\frac{\sqrt{3}}{2} = 21\sqrt{3}$
$A=21\sqrt{3}$. $\frac{1}{2}(12)(7)\frac{\sqrt{3}}{2} = 21\sqrt{3}$
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Find $A$ if $a=9$, $b=4$, and $C=45^\circ$ with $\sin(45^\circ)=\frac{\sqrt{2}}{2}$.
Find $A$ if $a=9$, $b=4$, and $C=45^\circ$ with $\sin(45^\circ)=\frac{\sqrt{2}}{2}$.
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$A=9\sqrt{2}$. $\frac{1}{2}(9)(4)\frac{\sqrt{2}}{2} = 9\sqrt{2}$
$A=9\sqrt{2}$. $\frac{1}{2}(9)(4)\frac{\sqrt{2}}{2} = 9\sqrt{2}$
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Find $A$ if $a=7$, $b=7$, and $C=120^\circ$ with $\sin(120^\circ)=\frac{\sqrt{3}}{2}$.
Find $A$ if $a=7$, $b=7$, and $C=120^\circ$ with $\sin(120^\circ)=\frac{\sqrt{3}}{2}$.
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$A=\frac{49\sqrt{3}}{4}$. $\frac{1}{2}(7)(7)\frac{\sqrt{3}}{2} = \frac{49\sqrt{3}}{4}$
$A=\frac{49\sqrt{3}}{4}$. $\frac{1}{2}(7)(7)\frac{\sqrt{3}}{2} = \frac{49\sqrt{3}}{4}$
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Find $A$ if $a=13$, $b=2$, and $C=0^\circ$.
Find $A$ if $a=13$, $b=2$, and $C=0^\circ$.
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$A=0$. Sine of zero makes the area zero (degenerate triangle).
$A=0$. Sine of zero makes the area zero (degenerate triangle).
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Find the height to base $a$ if $b=10$ and $C=30^\circ$.
Find the height to base $a$ if $b=10$ and $C=30^\circ$.
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$h=5$. $h = 10\sin(30°) = 10(\frac{1}{2}) = 5$
$h=5$. $h = 10\sin(30°) = 10(\frac{1}{2}) = 5$
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