Applying Laws of Sines and Cosines - Geometry
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What is the Law of Cosines formula for side $a$ in triangle $ABC$?
What is the Law of Cosines formula for side $a$ in triangle $ABC$?
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$a^2=b^2+c^2-2bc\cos A$. Relates side $a$ to the other two sides and the cosine of opposite angle $A$.
$a^2=b^2+c^2-2bc\cos A$. Relates side $a$ to the other two sides and the cosine of opposite angle $A$.
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What is the Law of Sines for triangle $ABC$ with sides $a$, $b$, $c$ opposite $A$, $B$, $C$?
What is the Law of Sines for triangle $ABC$ with sides $a$, $b$, $c$ opposite $A$, $B$, $C$?
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$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Each side divided by its opposite angle's sine equals the same ratio.
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Each side divided by its opposite angle's sine equals the same ratio.
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What is the rearranged Law of Cosines expression for $\cos A$?
What is the rearranged Law of Cosines expression for $\cos A$?
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$\cos A=\frac{b^2+c^2-a^2}{2bc}$. Isolates $\cos A$ by rearranging the Law of Cosines formula.
$\cos A=\frac{b^2+c^2-a^2}{2bc}$. Isolates $\cos A$ by rearranging the Law of Cosines formula.
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What is the rearranged Law of Cosines expression for $\cos B$?
What is the rearranged Law of Cosines expression for $\cos B$?
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$\cos B=\frac{a^2+c^2-b^2}{2ac}$. Isolates $\cos B$ by rearranging the Law of Cosines formula.
$\cos B=\frac{a^2+c^2-b^2}{2ac}$. Isolates $\cos B$ by rearranging the Law of Cosines formula.
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What is the Law of Cosines formula for side $c$ in triangle $ABC$?
What is the Law of Cosines formula for side $c$ in triangle $ABC$?
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$c^2=a^2+b^2-2ab\cos C$. Relates side $c$ to the other two sides and the cosine of opposite angle $C$.
$c^2=a^2+b^2-2ab\cos C$. Relates side $c$ to the other two sides and the cosine of opposite angle $C$.
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What is the Law of Cosines formula for side $b$ in triangle $ABC$?
What is the Law of Cosines formula for side $b$ in triangle $ABC$?
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$b^2=a^2+c^2-2ac\cos B$. Relates side $b$ to the other two sides and the cosine of opposite angle $B$.
$b^2=a^2+c^2-2ac\cos B$. Relates side $b$ to the other two sides and the cosine of opposite angle $B$.
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Find $C$ in degrees if $\cos C=0$.
Find $C$ in degrees if $\cos C=0$.
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$C=90^\circ$. Cosine equals zero when the angle is $90°$.
$C=90^\circ$. Cosine equals zero when the angle is $90°$.
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Which method should you use first for $SAS$ data: Law of Sines or Law of Cosines?
Which method should you use first for $SAS$ data: Law of Sines or Law of Cosines?
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Law of Cosines. Two sides and included angle require Law of Cosines first.
Law of Cosines. Two sides and included angle require Law of Cosines first.
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Which method should you use first for $ASA$ data: Law of Sines or Law of Cosines?
Which method should you use first for $ASA$ data: Law of Sines or Law of Cosines?
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Law of Sines. Two angles and one side can use Law of Sines directly.
Law of Sines. Two angles and one side can use Law of Sines directly.
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Which method should you use first for $SSS$ data: Law of Sines or Law of Cosines?
Which method should you use first for $SSS$ data: Law of Sines or Law of Cosines?
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Law of Cosines. Three sides require Law of Cosines to find any angle.
Law of Cosines. Three sides require Law of Cosines to find any angle.
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What is the Law of Sines proportion to solve for $A$ given $a$, $b$, and $B$?
What is the Law of Sines proportion to solve for $A$ given $a$, $b$, and $B$?
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$\sin A=\frac{a\sin B}{b}$. Rearranged Law of Sines to isolate $\sin A$.
$\sin A=\frac{a\sin B}{b}$. Rearranged Law of Sines to isolate $\sin A$.
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What is the possible second angle in the $SSA$ ambiguous case if one solution has angle $A$?
What is the possible second angle in the $SSA$ ambiguous case if one solution has angle $A$?
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$180^\circ-A$. Sine is positive in quadrants I and II, creating two possible angles.
$180^\circ-A$. Sine is positive in quadrants I and II, creating two possible angles.
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What is the condition for no triangle in $SSA$ when given $A$ acute, opposite side $a$, adjacent side $b$, and $a<b\sin A$?
What is the condition for no triangle in $SSA$ when given $A$ acute, opposite side $a$, adjacent side $b$, and $a<b\sin A$?
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No triangle. When $a < b\sin A$, the given side is too short to reach the base.
No triangle. When $a < b\sin A$, the given side is too short to reach the base.
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What is the area formula using two sides and the included angle, suitable with $SAS$ data?
What is the area formula using two sides and the included angle, suitable with $SAS$ data?
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$K=\frac{1}{2}bc\sin A$. Uses two sides and their included angle to find triangle area.
$K=\frac{1}{2}bc\sin A$. Uses two sides and their included angle to find triangle area.
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What is the ambiguous case associated with the Law of Sines called?
What is the ambiguous case associated with the Law of Sines called?
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$SSA$ ambiguous case. Two sides and non-included angle can create zero, one, or two triangles.
$SSA$ ambiguous case. Two sides and non-included angle can create zero, one, or two triangles.
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What is the condition for exactly one right triangle in $SSA$ when $A$ is acute and $a=b\sin A$?
What is the condition for exactly one right triangle in $SSA$ when $A$ is acute and $a=b\sin A$?
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One triangle (right). When $a = b\sin A$, the triangle forms exactly one right triangle.
One triangle (right). When $a = b\sin A$, the triangle forms exactly one right triangle.
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What is the condition for two triangles in $SSA$ when $A$ is acute and $b\sin A<a<b$?
What is the condition for two triangles in $SSA$ when $A$ is acute and $b\sin A<a<b$?
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Two triangles. The intermediate range creates two possible triangles (ambiguous case).
Two triangles. The intermediate range creates two possible triangles (ambiguous case).
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What is the condition for exactly one triangle in $SSA$ when $A$ is acute and $a\ge b$?
What is the condition for exactly one triangle in $SSA$ when $A$ is acute and $a\ge b$?
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One triangle. When the known side is longest, only one triangle is possible.
One triangle. When the known side is longest, only one triangle is possible.
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What is the condition for exactly one triangle in $SSA$ when $A$ is obtuse and $a>b$?
What is the condition for exactly one triangle in $SSA$ when $A$ is obtuse and $a>b$?
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One triangle. Obtuse angle with longer opposite side creates exactly one triangle.
One triangle. Obtuse angle with longer opposite side creates exactly one triangle.
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What is the condition for no triangle in $SSA$ when $A$ is obtuse and $a\le b$?
What is the condition for no triangle in $SSA$ when $A$ is obtuse and $a\le b$?
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No triangle. Obtuse angle requires its opposite side to be the longest.
No triangle. Obtuse angle requires its opposite side to be the longest.
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Identify the missing step: If $\sin A=\frac{a\sin B}{b}$, what must be checked before taking $A=\arcsin(\cdot)$?
Identify the missing step: If $\sin A=\frac{a\sin B}{b}$, what must be checked before taking $A=\arcsin(\cdot)$?
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Check for a second solution $180^\circ-A$ (ambiguous case). Must check if $180° - A$ also creates a valid triangle solution.
Check for a second solution $180^\circ-A$ (ambiguous case). Must check if $180° - A$ also creates a valid triangle solution.
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Find $K$ if $a=10$, $b=6$, and included angle $C=30^\circ$.
Find $K$ if $a=10$, $b=6$, and included angle $C=30^\circ$.
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$K=\frac{1}{2}\cdot 10\cdot 6\sin 30^\circ$. Apply area formula with sides $a$, $b$ and included angle $C$.
$K=\frac{1}{2}\cdot 10\cdot 6\sin 30^\circ$. Apply area formula with sides $a$, $b$ and included angle $C$.
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Which triangle data type is typically solved first using the Law of Cosines: $SSS$ or $AAS$?
Which triangle data type is typically solved first using the Law of Cosines: $SSS$ or $AAS$?
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$SSS$. Three sides require Law of Cosines to find any angle.
$SSS$. Three sides require Law of Cosines to find any angle.
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Find $\sin B$ if $a=9$, $A=50^\circ$, and $b=12$ using the Law of Sines.
Find $\sin B$ if $a=9$, $A=50^\circ$, and $b=12$ using the Law of Sines.
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$\sin B=\frac{12\sin 50^\circ}{9}$. Apply $\frac{\sin B}{b} = \frac{\sin A}{a}$ and solve for $\sin B$.
$\sin B=\frac{12\sin 50^\circ}{9}$. Apply $\frac{\sin B}{b} = \frac{\sin A}{a}$ and solve for $\sin B$.
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What is the sine definition used in a Law of Cosines proof for angle $A$ in the same setup with opposite height $h$ and hypotenuse $c$?
What is the sine definition used in a Law of Cosines proof for angle $A$ in the same setup with opposite height $h$ and hypotenuse $c$?
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$\sin A=\frac{h}{c}$. Opposite side $h$ over hypotenuse $c$ defines sine of angle $A$.
$\sin A=\frac{h}{c}$. Opposite side $h$ over hypotenuse $c$ defines sine of angle $A$.
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What is the cosine definition used in a Law of Cosines proof for angle $A$ in a right triangle with adjacent $x$ and hypotenuse $c$?
What is the cosine definition used in a Law of Cosines proof for angle $A$ in a right triangle with adjacent $x$ and hypotenuse $c$?
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$\cos A=\frac{x}{c}$. Adjacent side $x$ over hypotenuse $c$ defines cosine of angle $A$.
$\cos A=\frac{x}{c}$. Adjacent side $x$ over hypotenuse $c$ defines cosine of angle $A$.
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Identify the correct formula to find an angle from three sides $a$, $b$, $c$ (choose the one for $C$).
Identify the correct formula to find an angle from three sides $a$, $b$, $c$ (choose the one for $C$).
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$C=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)$. Inverse cosine of the rearranged Law of Cosines expression.
$C=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)$. Inverse cosine of the rearranged Law of Cosines expression.
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What equality follows immediately from $b\sin A=a\sin B$ in a Law of Sines proof?
What equality follows immediately from $b\sin A=a\sin B$ in a Law of Sines proof?
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$\frac{a}{\sin A}=\frac{b}{\sin B}$. Dividing both sides by $\sin A \sin B$ yields the Law of Sines ratio.
$\frac{a}{\sin A}=\frac{b}{\sin B}$. Dividing both sides by $\sin A \sin B$ yields the Law of Sines ratio.
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What is the height relation used in a common proof of the Law of Sines when dropping altitude $h$ to side $c$ from $C$?
What is the height relation used in a common proof of the Law of Sines when dropping altitude $h$ to side $c$ from $C$?
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$h=b\sin A=a\sin B$. Altitude creates two right triangles with the same height from vertex $C$.
$h=b\sin A=a\sin B$. Altitude creates two right triangles with the same height from vertex $C$.
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Find $a$ using the Law of Sines if $A=30^\circ$, $B=45^\circ$, and $b=10$.
Find $a$ using the Law of Sines if $A=30^\circ$, $B=45^\circ$, and $b=10$.
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$a=10\cdot\frac{\sin 30^\circ}{\sin 45^\circ}$. Apply $\frac{a}{\sin A} = \frac{b}{\sin B}$ and solve for $a$.
$a=10\cdot\frac{\sin 30^\circ}{\sin 45^\circ}$. Apply $\frac{a}{\sin A} = \frac{b}{\sin B}$ and solve for $a$.
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