All Circles Are Similar - Precalculus

1. Use the Law of Cosines: find $c$ if $a=5$, $b=5$, and included angle $C=60^\circ$.

   $c=\sqrt{5^2+5^2-2\cdot^5\cdot^5\cos 60^\circ}=5$. Isosceles with $60°$ angle forms equilateral triangle.

2. Find $A$ if $\sin A=\frac{1}{2}$ and $A$ is acute.

   $30^\circ$. $\sin 30°=\frac{1}{2}$ from special right triangle.

3. State the Law of Sines for triangle $ABC$ using sides $a,b,c$ opposite angles $A,B,C$.

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

4. In any triangle, which side is opposite angle $A$ when using standard notation?

   $a$. Standard notation pairs lowercase sides with uppercase angles.

5. Use the Law of Sines: if $a=10$, $A=30^\circ$, and $B=60^\circ$, what is $b$?

   $b=10\frac{\sin 60^\circ}{\sin 30^\circ}=10\sqrt{3}$. Apply $\frac{a}{\sin A}=\frac{b}{\sin B}$ and simplify.

6. Identify the included angle for sides $b$ and $c$ in triangle $ABC$ (standard opposite-side notation).

   $A$. Angle $A$ is between sides $b$ and $c$ in standard notation.

7. What is the triangle angle sum rule used after finding two angles?

   $A+B+C=180^\circ$. Sum of interior angles in any triangle equals $180°$.

8. Use the Law of Sines: if $a=12$, $A=90^\circ$, and $B=30^\circ$, what is $b$?

   $b=12\frac{\sin 30^\circ}{\sin 90^\circ}=6$. Apply sine ratio with $\sin 90^\circ=1$ and $\sin 30^\circ=\frac{1}{2}$.

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

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

10. Find the magnitude of the resultant of two forces $10$ and $10$ with included angle $60^\circ$.

    $R=\sqrt{10^2+10^2+2\cdot10\cdot10\cos60^\circ}=10\sqrt{3}$. Use cosine formula with angle between forces.

11. Which triangle data type is directly solvable using the Law of Sines without extra steps: $AAS$, $ASA$, $SAS$, or $SSS$?

    $AAS$ or $ASA$. Both have two angles and one side, perfect for sine ratio.

12. Use the Law of Sines: if $a=8$, $A=45^\circ$, and $B=30^\circ$, what is $b$ (exact form)?

    $b=8\frac{\sin 30^\circ}{\sin 45^\circ}=4\sqrt{2}$. Apply sine ratio and simplify using $\sin 30^\circ=\frac{1}{2}$, $\sin 45^\circ=\frac{\sqrt{2}}{2}$.

13. State the Law of Cosines formula for angle $A$ using sides $a,b,c$.

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

14. What triangle information type is the standard use case for the Law of Sines: $AAS, ASA, SAA, SSA,$ or $SSS$?

    $AAS, ASA, \text{or } SAA$. Two angles and a side allow unique triangle solution.

15. State the Law of Sines for a triangle with sides $a,b,c$ opposite angles $A,B,C$.

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

16. State the Law of Cosines formula that solves for side $a$ in triangle $ABC$.

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

17. Which triangle data type is the classic ambiguous case for the Law of Sines: $SSA$, $SAS$, or $SSS$?

    $SSA$. Two sides and non-included angle can yield 0, 1, or 2 triangles.

18. Classify the triangle by angles if $a=3$, $b=4$, and $c=5$ (largest side is $5$).

    $\text{Right, since }5^2=3^2+4^2$. Pythagorean theorem confirms $90°$ angle.

19. Find the missing angle if $A=35^\circ$ and $B=75^\circ$ in a triangle.

    $C=70^\circ$. Triangle angles sum to $180^\circ$.

20. State the Law of Cosines formula for side $a$ in terms of $b,c,$ and included angle $A$.

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

21. In the $SSA$ case, what is the first step to test for $0$, $1$, or $2$ triangles when angle $A$ is known?

    Compute $h=b\sin A$. Height $h$ determines if side $a$ can reach the opposite side.

22. In the $SSA$ case with acute $A$, what condition gives two triangles?

    $h<b$ and $h<a<b$ where $h=b\sin A$. Side $a$ can swing to two positions when $A$ is acute.

23. Use the Law of Sines: find $b$ if $A=45^\circ$, $B=30^\circ$, and $a=8$.

    $b=\frac{8\sin 30^\circ}{\sin 45^\circ}=4\sqrt{2}$. Calculate: $b=\frac{8(0.5)}{\frac{\sqrt{2}}{2}}=4\sqrt{2}$.

24. Find $C$ if $A=35^\circ$ and $B=65^\circ$ in a triangle.

    $80^\circ$. Apply angle sum: $C=180°-35°-65°=80°$.

25. In the $SSA$ case with given $A$, $a$, and $b$, what condition gives no triangle?

    $a<h=b\sin A$. Side $a$ is too short to reach the opposite side.

26. What triangle information type is the standard use case for the Law of Cosines: $SAS, SSS,$ or $ASA$?

    $SAS\text{ or }SSS$. Two sides with included angle or three sides determine triangle.

27. State the altitude relation for the $SSA$ case with known $A,a,$ and adjacent side $b$.

    $h=b\sin A$. Height from vertex to opposite side uses sine of angle.

28. For $SSA$ with $A=30^\circ$, $a=4$, $b=10$, how many triangles are possible?

    $0\text{ triangles}$. Since $a<b\sin A=5$, no triangle exists.

29. State the Law of Sines form that directly solves for $a$ given $A,b,$ and $B$.

    $a=\frac{b\sin A}{\sin B}$. Cross-multiply Law of Sines to isolate desired side.

30. State the Law of Cosines formula that solves for angle $A$ given sides $a,b,c$.

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

31. Identify the ambiguous case in triangle solving that can yield $0,1,$ or $2$ solutions.

    $SSA$. Two sides and non-included angle may have multiple solutions.

32. For $SSA$ with $A=30^\circ$, $a=6$, $b=10$, how many triangles are possible?

    $2\text{ triangles}$. Since $b\sin A<a<b$, two triangles possible.

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