Proving, Applying Laws of Sines/Cosines - Geometry

1. What is the valid angle range for an interior triangle angle $C$?

   $0^\circ<C<180^\circ$. Interior angles must be between 0° and 180°.

2. What area expression results from substituting $h=b\sin(C)$ into $A=\frac{1}{2}ah$?

   $A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.

3. Find and correct the error: $A=ab\sin(C)$ is used for triangle area.

   Correct: $A=\frac{1}{2}ab\sin(C)$. Missing the factor of $\frac{1}{2}$ in the area formula.

4. What is the area formula using sides $a,c$ and included angle $B$?

   $A_{\triangle}=\frac{1}{2}ac\sin(B)$. Uses sides $a$ and $c$ with their included angle $B$.

5. Find $A$ if $a=7$, $b=7$, and $C=120^\circ$ with $\sin(120^\circ)=\frac{\sqrt{3}}{2}$.

   $A=\frac{49\sqrt{3}}{4}$. $\frac{1}{2}(7)(7)\frac{\sqrt{3}}{2} = \frac{49\sqrt{3}}{4}$

6. What is $h$ after solving $\sin(C)=\frac{h}{b}$ for $h$?

   $h=b\sin(C)$. Multiply both sides by $b$ to isolate $h$.

7. Find $b$ if $A=24$, $a=8$, and $C=30^\circ$.

   $b=12$. $24 = \frac{1}{2}(8)(b)(\frac{1}{2})$, so $b = 12$

8. Find the height to base $a$ if $b=10$ and $C=30^\circ$.

   $h=5$. $h = 10\sin(30°) = 10(\frac{1}{2}) = 5$

9. What is $\sin(180^\circ-C)$ in terms of $\sin(C)$?

   $\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity: sine values are equal.

10. What is the height to base $a$ if the altitude splits off angle $C$ in a right triangle with hypotenuse $b$?

    $h=b\sin(C)$. Height is opposite to angle $C$ in the right triangle.

11. Find the height to base $a$ if $b=12$ and $C=60^\circ$ with $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.

    $h=6\sqrt{3}$. $h = 12\sin(60°) = 12(\frac{\sqrt{3}}{2}) = 6\sqrt{3}$

12. What construction handles the case when the altitude from a vertex falls outside the triangle?

    Extend the opposite side and drop a perpendicular to the extension. Handles obtuse triangles where altitude falls outside.

13. What is the area formula using sides $a,b$ and included angle $C$ (alternate notation)?

    $A_{\triangle}=\frac{1}{2}ab\sin(C)$. Uses sides $a$ and $b$ with their included angle $C$.

14. Identify the included angle for sides $a$ and $b$ in standard triangle notation.

    $C$. The angle between sides $a$ and $b$ is angle $C$.

15. Find $A$ if $a=10$, $b=6$, and $C=30^\circ$ using $A=\frac{1}{2}ab\sin(C)$.

    $A=15$. $\frac{1}{2}(10)(6)\sin(30°) = \frac{1}{2}(10)(6)(\frac{1}{2}) = 15$

16. Which trigonometric function appears in the two-sides-included-angle area formula?

    $\sin$. Sine relates the height to the sides and included angle.

17. Which sides are multiplied in $A=\frac{1}{2}ab\sin(C)$?

    The two sides that form angle $C$: $a$ and $b$. These are the sides that form the included angle $C$.

18. What right-triangle trig definition gives height when hypotenuse is $a$ and angle is $C$?

    $\sin(C)=\frac{h}{a}$. Sine equals opposite over hypotenuse in right triangles.

19. What is the standard base-height area formula for any triangle?

    $A=\frac{1}{2}bh$. Uses base times height, divided by 2.

20. Find $A$ if $a=12$, $b=7$, and $C=60^\circ$ with $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.

    $A=21\sqrt{3}$. $\frac{1}{2}(12)(7)\frac{\sqrt{3}}{2} = 21\sqrt{3}$

21. What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=0^\circ$?

    $A=0$. Sine of zero equals zero, making area zero (degenerate).

22. Find $A$ if $a=6$, $c=10$, and $B=90^\circ$ (angle between $a$ and $c$).

    $A_{\triangle}=30$. $\frac{1}{2}(6)(10)\sin(90°) = \frac{1}{2}(6)(10)(1) = 30$

23. What does the formula $A=\frac{1}{2}ab\sin(C)$ reduce to when $C=180^\circ$?

    $A=0$. Sine of 180° equals zero, making area zero (degenerate).

24. Which identity justifies using $\sin(C)$ even if the altitude meets an extension and forms angle $180^\circ-C$?

    $\sin(180^\circ-C)=\sin(C)$. Supplementary angle identity for sine function.

25. What does $h$ represent in the formula $A=\frac{1}{2}bh$?

    $h$ is the perpendicular height to base $b$. The perpendicular distance from vertex to base.

26. Find $\sin(C)$ if the altitude to base $a$ is $h=6$ and the adjacent side is $b=10$.

    $\sin(C)=\frac{3}{5}$. $\sin(C) = \frac{h}{b} = \frac{6}{10} = \frac{3}{5}$

27. Find $A$ if $a=14$, $b=9$, and $\sin(C)=\frac{1}{2}$.

    $A=\frac{63}{2}$. $\frac{1}{2}(14)(9)(\frac{1}{2}) = \frac{63}{2}$

28. Identify the height to side $a$ in terms of side $b$ and included angle $C$.

    $h=b\sin(C)$. From right triangle with hypotenuse $b$ and angle $C$.

29. What is $\sin(90^\circ)$, and what does it imply for area when $C=90^\circ$?

    $\sin(90^\circ)=1$, so $A=\frac{1}{2}ab$. Right angle makes formula reduce to standard rectangle area.

30. Find $A$ if $a=9$, $b=4$, and $C=45^\circ$ with $\sin(45^\circ)=\frac{\sqrt{2}}{2}$.

    $A=9\sqrt{2}$. $\frac{1}{2}(9)(4)\frac{\sqrt{2}}{2} = 9\sqrt{2}$

31. Find $\sin(C)$ if $A=30$, $a=10$, and $b=12$ using $A=\frac{1}{2}ab\sin(C)$.

    $\sin(C)=\frac{1}{2}$. $30 = \frac{1}{2}(10)(12)\sin(C)$, so $\sin(C) = \frac{1}{2}$

32. Which option is the correct area: $\frac{1}{2}ab\sin(C)$ or $\frac{1}{2}ab\sin(A)$, when $C$ is between $a$ and $b$?

    $\frac{1}{2}ab\sin(C)$. Must use the included angle $C$ between sides $a$ and $b$.

33. What auxiliary line is drawn to derive $A=\frac{1}{2}ab\sin(C)$?

    A perpendicular from a vertex to the opposite side (or its extension). Creates a right triangle to find height using trigonometry.

34. What right-triangle trig definition gives height when hypotenuse is $b$ and angle is $C$?

    $\sin(C)=\frac{h}{b}$. Sine equals opposite over hypotenuse in right triangles.

35. Find $A$ if $a=13$, $b=2$, and $C=0^\circ$.

    $A=0$. Sine of zero makes the area zero (degenerate triangle).

36. What is the area formula using two sides $a,b$ and included angle $C$?

    $A=\frac{1}{2}ab\sin(C)$. Uses two sides and the included angle between them.

37. What is the key right-triangle relationship between opposite leg, hypotenuse, and sine?

    $\text{opposite}=\text{hypotenuse}\cdot\sin(\text{angle})$. Basic trigonometric identity for right triangles.

38. Find $A$ if $b=11$, $c=7$, and $A=30^\circ$ (angle between $b$ and $c$).

    $A_{\triangle}=\frac{77}{4}$. $\frac{1}{2}(11)(7)\sin(30°) = \frac{1}{2}(11)(7)(\frac{1}{2}) = \frac{77}{4}$

39. What is the sign of $\sin(C)$ for an interior triangle angle $C$?

    $\sin(C)>0$ for $0^\circ<C<180^\circ$. Sine is positive in the first and second quadrants.

40. Which option is the correct height to base $a$: $b\sin(C)$ or $b\cos(C)$, when $C$ is between $a$ and $b$?

    $b\sin(C)$. Sine gives the opposite side (height) in the right triangle.

41. Identify the included angle for sides $b$ and $c$ in standard triangle notation.

    $A$. The angle between sides $b$ and $c$ is angle $A$.

42. Find $A$ if $a=8$, $b=5$, and $C=90^\circ$ using $A=\frac{1}{2}ab\sin(C)$.

    $A=20$. $\frac{1}{2}(8)(5)\sin(90°) = \frac{1}{2}(8)(5)(1) = 20$

43. Find and correct the error: $A=\frac{1}{2}ab\cos(C)$ is used for triangle area.

    Correct: $A=\frac{1}{2}ab\sin(C)$. Must use sine, not cosine, for triangle area formula.

44. Find $a$ if $A=35$, $b=10$, and $C=45^\circ$ with $\sin(45^\circ)=\frac{\sqrt{2}}{2}$.

    $a=7\sqrt{2}$. $35 = \frac{1}{2}(a)(10)(\frac{\sqrt{2}}{2})$, so $a = 7\sqrt{2}$

45. Identify the included angle for sides $a$ and $c$ in standard triangle notation.

    $B$. The angle between sides $a$ and $c$ is angle $B$.

46. Identify the height to side $b$ in terms of side $a$ and included angle $C$.

    $h=a\sin(C)$. From right triangle with hypotenuse $a$ and angle $C$.

47. In $A=\frac{1}{2}ab\sin(C)$, what does $C$ represent?

    The included angle between sides $a$ and $b$. The angle between the two given sides $a$ and $b$.

48. What is the area formula using sides $b,c$ and included angle $A$?

    $A_{\triangle}=\frac{1}{2}bc\sin(A)$. Uses sides $b$ and $c$ with their included angle $A$.

49. Find and correct the error: $h=b\cos(C)$ is used as the altitude to base $a$.

    Correct: $h=b\sin(C)$. Height uses sine, not cosine, for the opposite side.

50. What area expression results from substituting $h=a\sin(C)$ into $A=\frac{1}{2}bh$?

    $A=\frac{1}{2}ab\sin(C)$. Substitute the height expression into base-height formula.

51. What is $h$ after solving $\sin(C)=\frac{h}{a}$ for $h$?

    $h=a\sin(C)$. Multiply both sides by $a$ to isolate $h$.

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