Applying Laws of Sines and Cosines - Geometry

1. Identify the correct first step to solve $SSA$ data: use Law of Sines to find an angle or Law of Cosines to find a side?

   Use Law of Sines to find an angle. With two sides and non-included angle, find the angle opposite the known side.

2. What is the circumradius form of the Law of Sines relating $a$ and $\sin A$?

   $\frac{a}{\sin A}=2R$. Relates any side-angle pair to twice the circumradius $R$.

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

   $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$. Extended Law of Sines including the circumradius relationship.

4. Identify the missing step: If $\sin A=\frac{a\sin B}{b}$, what must be checked before taking $A=\arcsin(\cdot)$?

   Check for a second solution $180^\circ-A$ (ambiguous case). Must check if $180° - A$ also creates a valid triangle solution.

5. Which method should you use first for $SSS$ data: Law of Sines or Law of Cosines?

   Law of Cosines. Three sides require Law of Cosines to find any angle.

6. Find $K$ if $a=10$, $b=6$, and included angle $C=30^\circ$.

   $K=\frac{1}{2}\cdot 10\cdot 6\sin 30^\circ$. Apply area formula with sides $a$, $b$ and included angle $C$.

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

   $c^2=a^2+b^2-2ab\cos C$. Relates side $c$ to the other two sides and the cosine of opposite angle $C$.

8. Which method should you use first for $ASA$ data: Law of Sines or Law of Cosines?

   Law of Sines. Two angles and one side can use Law of Sines directly.

9. What is the Pythagorean Theorem statement that the Law of Cosines generalizes?

   $a^2=b^2+c^2$ when $A=90^\circ$. The Pythagorean Theorem is a special case when one angle is $90°$.

10. What is the possible second angle in the $SSA$ ambiguous case if one solution has angle $A$?

    $180^\circ-A$. Sine is positive in quadrants I and II, creating two possible angles.

11. Which method should you use first for $SAS$ data: Law of Sines or Law of Cosines?

    Law of Cosines. Two sides and included angle require Law of Cosines first.

12. Which triangle data type is typically solved first using the Law of Cosines: $SSS$ or $AAS$?

    $SSS$. Three sides require Law of Cosines to find any angle.

13. Find $C$ in degrees if $\cos C=0$.

    $C=90^\circ$. Cosine equals zero when the angle is $90°$.

14. What is the condition for exactly one right triangle in $SSA$ when $A$ is acute and $a=b\sin A$?

    One triangle (right). When $a = b\sin A$, the triangle forms exactly one right triangle.

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

    $b^2=a^2+c^2-2ac\cos B$. Relates side $b$ to the other two sides and the cosine of opposite angle $B$.

16. What is the condition for no triangle in $SSA$ when $A$ is obtuse and $a\le b$?

    No triangle. Obtuse angle requires its opposite side to be the longest.

17. 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 $c$?

    $\sin A=\frac{h}{c}$. Opposite side $h$ over hypotenuse $c$ defines sine of angle $A$.

18. What is the height relation used in a common proof of the Law of Sines when dropping altitude $h$ to side $c$ from $C$?

    $h=b\sin A=a\sin B$. Altitude creates two right triangles with the same height from vertex $C$.

19. Find $c$ using the Law of Sines if $C=60^\circ$, $A=45^\circ$, and $a=8$.

    $c=8\cdot\frac{\sin 60^\circ}{\sin 45^\circ}$. Apply $\frac{c}{\sin C} = \frac{a}{\sin A}$ and solve for $c$.

20. Find $R$ if $a=12$ and $A=30^\circ$ using $\frac{a}{\sin A}=2R$.

    $R=\frac{12}{2\sin 30^\circ}$. Use circumradius formula $R = \frac{a}{2\sin A}$.

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

    $a^2=b^2+c^2-2bc\cos A$. Relates side $a$ to the other two sides and the cosine of opposite angle $A$.

22. What is the ambiguous case associated with the Law of Sines called?

    $SSA$ ambiguous case. Two sides and non-included angle can create zero, one, or two triangles.

23. Find $c$ with the Law of Cosines if $a=5$, $b=12$, and $C=90^\circ$.

    $c=\sqrt{5^2+12^2}$. When $C = 90°$, Law of Cosines reduces to Pythagorean Theorem.

24. What is the condition for exactly one triangle in $SSA$ when $A$ is obtuse and $a>b$?

    One triangle. Obtuse angle with longer opposite side creates exactly one triangle.

25. Identify the correct formula to find an angle from three sides $a$, $b$, $c$ (choose the one for $C$).

    $C=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)$. Inverse cosine of the rearranged Law of Cosines expression.

26. Find $B$ if $A=40^\circ$ and $C=65^\circ$ in triangle $ABC$.

    $B=75^\circ$. Angles in a triangle sum to $180°$: $B = 180° - 40° - 65°$.

27. Which triangle data type is typically solved first using the Law of Cosines: $SAS$ or $ASA$?

    $SAS$. Two sides and included angle require Law of Cosines to find the third side.

28. What equality follows immediately from $b\sin A=a\sin B$ in a Law of Sines proof?

    $\frac{a}{\sin A}=\frac{b}{\sin B}$. Dividing both sides by $\sin A \sin B$ yields the Law of Sines ratio.

29. What is the cosine definition used in a Law of Cosines proof for angle $A$ in a right triangle with adjacent $x$ and hypotenuse $c$?

    $\cos A=\frac{x}{c}$. Adjacent side $x$ over hypotenuse $c$ defines cosine of angle $A$.

30. What is the condition for no triangle in $SSA$ when given $A$ acute, opposite side $a$, adjacent side $b$, and $a<b\sin A$?

    No triangle. When $a < b\sin A$, the given side is too short to reach the base.

31. What is the rearranged Law of Cosines expression for $\cos C$?

    $\cos C=\frac{a^2+b^2-c^2}{2ab}$. Isolates $\cos C$ by rearranging the Law of Cosines formula.

32. Find and correct the mismatch: $\frac{a}{\sin B}=\frac{b}{\sin A}$ in triangle $ABC$.

    Correct: $\frac{a}{\sin A}=\frac{b}{\sin B}$. Sides must pair with their opposite angles in the Law of Sines.

33. What is the Law of Sines for triangle $ABC$ with sides $a$, $b$, $c$ opposite $A$, $B$, $C$?

    $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Each side divided by its opposite angle's sine equals the same ratio.

34. What is the area formula using two sides and the included angle, suitable with $SAS$ data?

    $K=\frac{1}{2}bc\sin A$. Uses two sides and their included angle to find triangle area.

35. What is the condition for exactly one triangle in $SSA$ when $A$ is acute and $a\ge b$?

    One triangle. When the known side is longest, only one triangle is possible.

36. What is the rearranged Law of Cosines expression for $\cos B$?

    $\cos B=\frac{a^2+c^2-b^2}{2ac}$. Isolates $\cos B$ by rearranging the Law of Cosines formula.

37. Which triangle data type is always solvable using the Law of Sines without ambiguity: $ASA$, $AAS$, $SAS$, or $SSS$?

    $ASA$ or $AAS$. Both cases provide two angles and one side, giving unique solutions.

38. Find $a$ using the Law of Sines if $A=30^\circ$, $B=45^\circ$, and $b=10$.

    $a=10\cdot\frac{\sin 30^\circ}{\sin 45^\circ}$. Apply $\frac{a}{\sin A} = \frac{b}{\sin B}$ and solve for $a$.

39. What is the rearranged Law of Cosines expression for $\cos A$?

    $\cos A=\frac{b^2+c^2-a^2}{2bc}$. Isolates $\cos A$ by rearranging the Law of Cosines formula.

40. Find $b$ using the Law of Sines if $A=35^\circ$, $B=70^\circ$, and $a=4$.

    $b=4\cdot\frac{\sin 70^\circ}{\sin 35^\circ}$. Apply $\frac{b}{\sin B} = \frac{a}{\sin A}$ and solve for $b$.

41. What is the area formula $K$ in terms of $a$, $c$, and included angle $B$?

    $K=\frac{1}{2}ac\sin B$. Area formula using sides $a$, $c$ and their included angle $B$.

42. Find $a^2$ using the Law of Cosines if $b=3$, $c=4$, and $A=120^\circ$.

    $a^2=3^2+4^2-2\cdot 3\cdot 4\cos 120^\circ$. Apply Law of Cosines with $\cos 120° = -\frac{1}{2}$.

43. Find $a$ with the Law of Cosines if $b=7$, $c=9$, and $A=60^\circ$.

    $a=\sqrt{7^2+9^2-2\cdot 7\cdot 9\cos 60^\circ}$. Apply Law of Cosines with known sides $b$, $c$ and included angle $A$.

44. What is the condition for two triangles in $SSA$ when $A$ is acute and $b\sin A<a<b$?

    Two triangles. The intermediate range creates two possible triangles (ambiguous case).

45. Identify the correct statement for side-angle correspondence in triangle $ABC$.

    $a\leftrightarrow A$, $b\leftrightarrow B$, $c\leftrightarrow C$. Standard notation: lowercase sides opposite uppercase angles.

46. What is the area formula $K$ in terms of $a$, $b$, and included angle $C$?

    $K=\frac{1}{2}ab\sin C$. Area formula using sides $a$, $b$ and their included angle $C$.

47. What does the Law of Cosines become when $A=180^\circ$ in $a^2=b^2+c^2-2bc\cos A$?

    $a^2=(b+c)^2$. When $\cos 180° = -1$, the formula becomes $a^2 = b^2 + c^2 + 2bc = (b+c)^2$.

48. Find $\sin B$ if $a=9$, $A=50^\circ$, and $b=12$ using the Law of Sines.

    $\sin B=\frac{12\sin 50^\circ}{9}$. Apply $\frac{\sin B}{b} = \frac{\sin A}{a}$ and solve for $\sin B$.

49. What does the Law of Cosines become when $A=0^\circ$ in the expression $a^2=b^2+c^2-2bc\cos A$?

    $a^2=(b-c)^2$. When $\cos 0° = 1$, the formula becomes $a^2 = b^2 + c^2 - 2bc = (b-c)^2$.

50. What does the Law of Cosines simplify to when $A=90^\circ$?

    $a^2=b^2+c^2$. When $\cos 90° = 0$, the $-2bc\cos A$ term vanishes.

51. What is the Law of Sines proportion to solve for $A$ given $a$, $b$, and $B$?

    $\sin A=\frac{a\sin B}{b}$. Rearranged Law of Sines to isolate $\sin A$.

52. Find $\cos C$ if $a=6$, $b=8$, and $c=10$.

    $\cos C=\frac{6^2+8^2-10^2}{2\cdot 6\cdot 8}$. Apply rearranged Law of Cosines to find $\cos C$ from three sides.

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