Applying Laws of Sines and Cosines - Precalculus

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

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

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

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

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

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

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

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

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

   $c=\sqrt{5^2+7^2-2(5)(7)\cos 60^\circ}$. Apply $c^2=a^2+b^2-2ab\cos C$ with given values.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates ratios of sides to sines of opposite angles in any triangle.

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

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

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

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

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

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

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

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

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