Applying Laws of Sines and Cosines - Pre-Calculus
Card 1 of 30
What is the angle-sum equation used after finding two angles in a triangle?
What is the angle-sum equation used after finding two angles in a triangle?
Tap to reveal answer
$A+B+C=180^\circ$. Triangle angles always sum to $180°$ or $\pi$ radians.
$A+B+C=180^\circ$. Triangle angles always sum to $180°$ or $\pi$ radians.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is obtuse and $a\le b$.
Identify the number of triangles in $SSA$ when $A$ is obtuse and $a\le b$.
Tap to reveal answer
$0$ triangles. Obtuse angle with shorter opposite side is impossible.
$0$ triangles. Obtuse angle with shorter opposite side is impossible.
← Didn't Know|Knew It →
Find $a$ using the Law of Sines: $A=30^\circ$, $B=60^\circ$, and $b=10$.
Find $a$ using the Law of Sines: $A=30^\circ$, $B=60^\circ$, and $b=10$.
Tap to reveal answer
$a=\frac{10\sin 30^\circ}{\sin 60^\circ}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $a$.
$a=\frac{10\sin 30^\circ}{\sin 60^\circ}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $a$.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is acute and $a\ge b$ (with $h=b\sin A$).
Identify the number of triangles in $SSA$ when $A$ is acute and $a\ge b$ (with $h=b\sin A$).
Tap to reveal answer
$1$ triangle. Side $a$ long enough to reach only one position.
$1$ triangle. Side $a$ long enough to reach only one position.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is acute and $h<a<b$ (with $h=b\sin A$).
Identify the number of triangles in $SSA$ when $A$ is acute and $h<a<b$ (with $h=b\sin A$).
Tap to reveal answer
$2$ triangles. Side $a$ can swing to two positions between $h$ and $b$.
$2$ triangles. Side $a$ can swing to two positions between $h$ and $b$.
← Didn't Know|Knew It →
Find $\cos C$ from sides using the Law of Cosines: sides $a$, $b$, $c$ with angle $C$ opposite $c$.
Find $\cos C$ from sides using the Law of Cosines: sides $a$, $b$, $c$ with angle $C$ opposite $c$.
Tap to reveal answer
$\cos C=\frac{a^2+b^2-c^2}{2ab}$. Rearrange Law of Cosines to isolate $\cos C$.
$\cos C=\frac{a^2+b^2-c^2}{2ab}$. Rearrange Law of Cosines to isolate $\cos C$.
← Didn't Know|Knew It →
Find $\sin B$ using the Law of Sines: $a=8$, $A=40^\circ$, and $b=10$.
Find $\sin B$ using the Law of Sines: $a=8$, $A=40^\circ$, and $b=10$.
Tap to reveal answer
$\sin B=\frac{10\sin 40^\circ}{8}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $\sin B$.
$\sin B=\frac{10\sin 40^\circ}{8}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $\sin B$.
← Didn't Know|Knew It →
Identify the correct step to start proving the Law of Sines: drop an altitude from which vertex?
Identify the correct step to start proving the Law of Sines: drop an altitude from which vertex?
Tap to reveal answer
Drop an altitude from a vertex to create two right triangles. Any vertex works; altitude creates right triangles for sine ratios.
Drop an altitude from a vertex to create two right triangles. Any vertex works; altitude creates right triangles for sine ratios.
← Didn't Know|Knew It →
Identify the key construction to start proving the Law of Cosines: what segment is drawn in triangle $ABC$?
Identify the key construction to start proving the Law of Cosines: what segment is drawn in triangle $ABC$?
Tap to reveal answer
Draw an altitude to form a right triangle and apply the Pythagorean Theorem. Altitude allows coordinate geometry or Pythagorean approach.
Draw an altitude to form a right triangle and apply the Pythagorean Theorem. Altitude allows coordinate geometry or Pythagorean approach.
← Didn't Know|Knew It →
Find $c$ using the Law of Cosines: $a=5$, $b=7$, and $C=60^\circ$.
Find $c$ using the Law of Cosines: $a=5$, $b=7$, and $C=60^\circ$.
Tap to reveal answer
$c=\sqrt{5^2+7^2-2(5)(7)\cos 60^\circ}$. Apply $c^2=a^2+b^2-2ab\cos C$ with given values.
$c=\sqrt{5^2+7^2-2(5)(7)\cos 60^\circ}$. Apply $c^2=a^2+b^2-2ab\cos C$ with given values.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is obtuse and $a>b$.
Identify the number of triangles in $SSA$ when $A$ is obtuse and $a>b$.
Tap to reveal answer
$1$ triangle. Obtuse angle requires longest opposite side.
$1$ triangle. Obtuse angle requires longest opposite side.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is acute and $a=h$ (with $h=b\sin A$).
Identify the number of triangles in $SSA$ when $A$ is acute and $a=h$ (with $h=b\sin A$).
Tap to reveal answer
$1$ triangle (right triangle). Side $a$ exactly reaches to form $90°$ angle.
$1$ triangle (right triangle). Side $a$ exactly reaches to form $90°$ angle.
← Didn't Know|Knew It →
Identify the number of triangles in $SSA$ when $A$ is acute and $a<h$ (with $h=b\sin A$).
Identify the number of triangles in $SSA$ when $A$ is acute and $a<h$ (with $h=b\sin A$).
Tap to reveal answer
$0$ triangles. Side $a$ too short to reach the opposite side.
$0$ triangles. Side $a$ too short to reach the opposite side.
← Didn't Know|Knew It →
State the area formula of a triangle using two sides $b$, $c$ and included angle $A$.
State the area formula of a triangle using two sides $b$, $c$ and included angle $A$.
Tap to reveal answer
$K=\frac{1}{2}bc\sin A$. Half the product of two sides times sine of included angle.
$K=\frac{1}{2}bc\sin A$. Half the product of two sides times sine of included angle.
← Didn't Know|Knew It →
Identify when the Law of Cosines is most directly applicable: $SSS$, $SAS$, $ASA$, or $AAS$.
Identify when the Law of Cosines is most directly applicable: $SSS$, $SAS$, $ASA$, or $AAS$.
Tap to reveal answer
$SSS$ or $SAS$. Use when you have all sides or two sides with included angle.
$SSS$ or $SAS$. Use when you have all sides or two sides with included angle.
← Didn't Know|Knew It →
Identify when the Law of Sines is most directly applicable: $ASA$, $AAS$, $SSA$, or $SAS$.
Identify when the Law of Sines is most directly applicable: $ASA$, $AAS$, $SSA$, or $SAS$.
Tap to reveal answer
$ASA$, $AAS$, or $SSA$. Use when you have angles and need to find sides.
$ASA$, $AAS$, or $SSA$. Use when you have angles and need to find sides.
← Didn't Know|Knew It →
State the Law of Cosines formula for side $c$ in triangle $ABC$.
State the Law of Cosines formula for side $c$ in triangle $ABC$.
Tap to reveal answer
$c^2=a^2+b^2-2ab\cos C$. Completes the set with angle $C$ opposite side $c$.
$c^2=a^2+b^2-2ab\cos C$. Completes the set with angle $C$ opposite side $c$.
← Didn't Know|Knew It →
State the Law of Cosines formula for side $b$ in triangle $ABC$.
State the Law of Cosines formula for side $b$ in triangle $ABC$.
Tap to reveal answer
$b^2=a^2+c^2-2ac\cos B$. Same pattern as for side $a$, with angle $B$ opposite side $b$.
$b^2=a^2+c^2-2ac\cos B$. Same pattern as for side $a$, with angle $B$ opposite side $b$.
← Didn't Know|Knew It →
State the Law of Cosines formula for side $a$ in triangle $ABC$.
State the Law of Cosines formula for side $a$ in triangle $ABC$.
Tap to reveal answer
$a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine term for non-right triangles.
$a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine term for non-right triangles.
← Didn't Know|Knew It →
State the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$.
State the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$.
Tap to reveal answer
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates ratios of sides to sines of opposite angles in any triangle.
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates ratios of sides to sines of opposite angles in any triangle.
← Didn't Know|Knew It →
What is the Law of Cosines formula for angle $A$ in terms of $a$, $b$, and $c$?
What is the Law of Cosines formula for angle $A$ in terms of $a$, $b$, and $c$?
Tap to reveal answer
$\cos A=\frac{b^2+c^2-a^2}{2bc}$. Rearranges Law of Cosines to solve for angle.
$\cos A=\frac{b^2+c^2-a^2}{2bc}$. Rearranges Law of Cosines to solve for angle.
← Didn't Know|Knew It →
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 $ABC$?
Tap to reveal answer
$c^2=a^2+b^2-2ab\cos C$. Same pattern with $c$ as the subject side.
$c^2=a^2+b^2-2ab\cos C$. Same pattern with $c$ as the subject side.
← Didn't Know|Knew It →
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 $ABC$?
Tap to reveal answer
$b^2=a^2+c^2-2ac\cos B$. Same pattern with $b$ as the subject side.
$b^2=a^2+c^2-2ac\cos B$. Same pattern with $b$ as the subject side.
← Didn't Know|Knew It →
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 $ABC$?
Tap to reveal answer
$a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine correction term.
$a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine correction term.
← Didn't Know|Knew It →
Which triangle data type typically indicates using the Law of Cosines: $AAS$, $ASA$, $SSS$, or $SAS$?
Which triangle data type typically indicates using the Law of Cosines: $AAS$, $ASA$, $SSS$, or $SAS$?
Tap to reveal answer
$SSS$ or $SAS$. All sides or two sides with included angle need cosine law.
$SSS$ or $SAS$. All sides or two sides with included angle need cosine law.
← Didn't Know|Knew It →
What is the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$?
What is the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$?
Tap to reveal answer
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates each side to the sine of its opposite angle.
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates each side to the sine of its opposite angle.
← Didn't Know|Knew It →
Identify the correct method: Given $a$, $b$, and included angle $C$, which law should you use first?
Identify the correct method: Given $a$, $b$, and included angle $C$, which law should you use first?
Tap to reveal answer
Use the Law of Cosines first. SAS requires cosine law to find third side first.
Use the Law of Cosines first. SAS requires cosine law to find third side first.
← Didn't Know|Knew It →
Find $B$ if $A=40^\circ$, $a=8$, and $b=10$ using the Law of Sines.
Find $B$ if $A=40^\circ$, $a=8$, and $b=10$ using the Law of Sines.
Tap to reveal answer
$B=\sin^{-1}!\left(\frac{10\sin 40^\circ}{8}\right)$. SSA case: use sine law but check if $\sin B > 1$.
$B=\sin^{-1}!\left(\frac{10\sin 40^\circ}{8}\right)$. SSA case: use sine law but check if $\sin B > 1$.
← Didn't Know|Knew It →
Identify the one-triangle condition in $SSA$ with acute $A$: what inequality using $a$ and $b$ guarantees $1$ solution?
Identify the one-triangle condition in $SSA$ with acute $A$: what inequality using $a$ and $b$ guarantees $1$ solution?
Tap to reveal answer
$a\ge b$. Longer opposite side to given angle ensures unique triangle.
$a\ge b$. Longer opposite side to given angle ensures unique triangle.
← Didn't Know|Knew It →
Identify the two-triangle condition in $SSA$ with acute $A$: what inequality using $a$, $b$, and $\sin A$ gives $2$ solutions?
Identify the two-triangle condition in $SSA$ with acute $A$: what inequality using $a$, $b$, and $\sin A$ gives $2$ solutions?
Tap to reveal answer
$b\sin A<a<b$. Side $a$ can swing to two positions, creating two triangles.
$b\sin A<a<b$. Side $a$ can swing to two positions, creating two triangles.
← Didn't Know|Knew It →