Precalculus Flashcards: Applying Laws Of Sines And Cosines

What this deck covers

This deck focuses on Applying Laws Of Sines And Cosines, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the two-triangle condition in $SSA$ with acute $A$: what inequality using $a$, $b$, and $\sin A$ gives $2$ solutions?

Answer: $b\sin A<a<b$. Side $a$ can swing to two positions, creating two triangles.

Flashcard 2: Identify the one-triangle condition in $SSA$ with acute $A$: what inequality using $a$ and $b$ guarantees $1$ solution?

Answer: $a\ge b$. Longer opposite side to given angle ensures unique triangle.

Flashcard 3: Identify when the Law of Cosines is most directly applicable: $SSS$, $SAS$, $ASA$, or $AAS$.

Answer: $SSS$ or $SAS$. Use when you have all sides or two sides with included angle.

Flashcard 4: Identify when the Law of Sines is most directly applicable: $ASA$, $AAS$, $SSA$, or $SAS$.

Answer: $ASA$, $AAS$, or $SSA$. Use when you have angles and need to find sides.

Flashcard 5: Find $c$ using the Law of Cosines: $a=5$, $b=7$, and $C=60^\circ$.

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.

Flashcard 6: Identify the number of triangles in $SSA$ when $A$ is acute and $a=h$ (with $h=b\sin A$).

Answer: $1$ triangle (right triangle). Side $a$ exactly reaches to form $90°$ angle.

Flashcard 7: Which triangle data type typically indicates using the Law of Cosines: $AAS$, $ASA$, $SSS$, or $SAS$?

Answer: $SSS$ or $SAS$. All sides or two sides with included angle need cosine law.

Flashcard 8: Identify the number of triangles in $SSA$ when $A$ is acute and $a\ge b$ (with $h=b\sin A$).

Answer: $1$ triangle. Side $a$ long enough to reach only one position.

Flashcard 9: State the Law of Cosines formula for side $b$ in triangle $ABC$.

Answer: $b^2=a^2+c^2-2ac\cos B$. Same pattern as for side $a$, with angle $B$ opposite side $b$.

Flashcard 10: Identify the number of triangles in $SSA$ when $A$ is acute and $a<h$ (with $h=b\sin A$).

Answer: $0$ triangles. Side $a$ too short to reach the opposite side.

Flashcard 11: What is the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$?

Answer: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates each side to the sine of its opposite angle.

Flashcard 12: What is the Law of Cosines formula for angle $A$ in terms of $a$, $b$, and $c$?

Answer: $\cos A=\frac{b^2+c^2-a^2}{2bc}$. Rearranges Law of Cosines to solve for angle.

Flashcard 13: What is the circumradius form of the Law of Sines for triangle $ABC$?

Answer: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$. $R$ is the circumradius of the triangle.

Flashcard 14: State the Law of Cosines formula for side $c$ in triangle $ABC$.

Answer: $c^2=a^2+b^2-2ab\cos C$. Completes the set with angle $C$ opposite side $c$.

Flashcard 15: What is the angle-sum equation used after finding two angles in a triangle?

Answer: $A+B+C=180^\circ$. Triangle angles always sum to $180°$ or $\pi$ radians.

Flashcard 16: Find $a$ using the Law of Sines: $A=30^\circ$, $B=60^\circ$, and $b=10$.

Answer: $a=\frac{10\sin 30^\circ}{\sin 60^\circ}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $a$.

Flashcard 17: Identify the key construction to start proving the Law of Cosines: what segment is drawn in triangle $ABC$?

Answer: Draw an altitude to form a right triangle and apply the Pythagorean Theorem. Altitude allows coordinate geometry or Pythagorean approach.

Flashcard 18: What is the area formula that helps prove the Law of Sines using two sides and an included angle?

Answer: $K=\frac{1}{2}bc\sin A$. Area equals half the product of two sides times sine of included angle.

Flashcard 19: Find $\sin B$ using the Law of Sines: $a=8$, $A=40^\circ$, and $b=10$.

Answer: $\sin B=\frac{10\sin 40^\circ}{8}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and solve for $\sin B$.

Flashcard 20: Which triangle data type typically indicates using the Law of Sines: $AAS$, $ASA$, $SSS$, or $SAS$?

Answer: $ASA$ or $AAS$. Two angles known means sine ratios work best.

Flashcard 21: State the Law of Cosines formula for side $a$ in triangle $ABC$.

Answer: $a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine term for non-right triangles.

Flashcard 22: What is the ambiguous case condition for possible two solutions when using the Law of Sines?

Answer: $SSA$ data with an acute given angle. Acute angle with two sides can yield 0, 1, or 2 triangles.

Flashcard 23: Identify the no-triangle condition in $SSA$ with acute $A$: what inequality using $a$, $b$, and $\sin A$ gives $0$ solutions?

Answer: $a<b\sin A$. Side $a$ too short to reach from $B$ to form triangle.

Flashcard 24: Find $\cos C$ from sides using the Law of Cosines: sides $a$, $b$, $c$ with angle $C$ opposite $c$.

Answer: $\cos C=\frac{a^2+b^2-c^2}{2ab}$. Rearrange Law of Cosines to isolate $\cos C$.

Flashcard 25: Identify the correct step to start proving the Law of Sines: drop an altitude from which vertex?

Answer: Drop an altitude from a vertex to create two right triangles. Any vertex works; altitude creates right triangles for sine ratios.

Flashcard 26: State the area formula of a triangle using two sides $b$, $c$ and included angle $A$.

Answer: $K=\frac{1}{2}bc\sin A$. Half the product of two sides times sine of included angle.

Flashcard 27: Identify the correct method: Given $a$, $b$, and included angle $C$, which law should you use first?

Answer: Use the Law of Cosines first. SAS requires cosine law to find third side first.

Flashcard 28: Find $B$ if $A=40^\circ$, $a=8$, and $b=10$ using the Law of Sines.

Answer: $B=\sin^{-1}\!\left(\frac{10\sin 40^\circ}{8}\right)$. SSA case: use sine law but check if $\sin B > 1$.

Flashcard 29: Identify the number of triangles in $SSA$ when $A$ is acute and $h<a<b$ (with $h=b\sin A$).

Answer: $2$ triangles. Side $a$ can swing to two positions between $h$ and $b$.

Flashcard 30: Identify the number of triangles in $SSA$ when $A$ is obtuse and $a>b$.

Answer: $1$ triangle. Obtuse angle requires longest opposite side.

Flashcard 31: What is the Law of Cosines formula for side $b$ in triangle $ABC$?

Answer: $b^2=a^2+c^2-2ac\cos B$. Same pattern with $b$ as the subject side.

Flashcard 32: State the Law of Sines for triangle $ABC$ with opposite sides $a$, $b$, and $c$.

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.

Flashcard 33: Identify the number of triangles in $SSA$ when $A$ is obtuse and $a\le b$.

Answer: $0$ triangles. Obtuse angle with shorter opposite side is impossible.

Flashcard 34: What is the Law of Cosines formula for side $a$ in triangle $ABC$?

Answer: $a^2=b^2+c^2-2bc\cos A$. Generalizes Pythagorean theorem with cosine correction term.

Flashcard 35: What is the Law of Cosines formula for side $c$ in triangle $ABC$?

Answer: $c^2=a^2+b^2-2ab\cos C$. Same pattern with $c$ as the subject side.

Flashcard 36: Identify the right-triangle condition in $SSA$ with acute $A$: what equality using $a$, $b$, and $\sin A$ gives $1$ solution?

Answer: $a=b\sin A$. Side $a$ exactly reaches to form right angle at $C$.