Precalculus Applications of Trigonometry and Vectors Syllabus

Applications of Trigonometry and Vectors syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Applications of Trigonometry and Vectors in Precalculus, not a summary of it.

1. 1

   Law of Sines

   3 topics

   • Solving Oblique Triangles (AAS, ASA)
   • The Ambiguous Case (SSA)
   • Area of an Oblique Triangle
2. 2

   Law of Cosines

   2 topics

   • Solving Triangles (SSS, SAS)
   • Heron's Area Formula
3. 3

   Vectors in the Plane

   3 topics

   • Component Form and Magnitude
   • Vector Operations
   • Applications of Vectors
4. 4

   Dot Product

   3 topics

   • Definition and Properties
   • Angle Between Vectors
   • Projections and Work
5. 5

   Complex Numbers in Trigonometric Form

   3 topics

   • Trigonometric (Polar) Form
   • Products and Quotients
   • DeMoivre's Theorem and nth Roots

Applications of Trigonometry and Vectors flashcards for Precalculus

23 of 50 cards from the Applications of Trigonometry and Vectors deck — real questions with worked answers.

1. What is the Law of Sines?

   For any triangle with sides $a,b,c$ opposite angles $A,B,C$: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

2. Which cases of a triangle (given information) are solved using the Law of Sines?

   The cases where you know an angle and its opposite side: AAS, ASA, and the ambiguous SSA case.

3. How do you solve an AAS or ASA triangle?

   Find the third angle using $A+B+C=180^{\circ}$, then apply the Law of Sines $\frac{a}{\sin A}=\frac{b}{\sin B}$ to find each remaining side.

4. What does the SSA configuration mean, and why is it called the 'ambiguous case'?

   SSA means two sides and a non-included angle are given. It is ambiguous because such data can produce zero, one, or two valid triangles.

5. In the SSA ambiguous case with given angle $A$, side $a$ (opposite $A$), and side $b$, how many triangles result when $A$ is acute?

   Let $h=b\sin A$. If $a<h$: no triangle; if $a=h$: one right triangle; if $h<a<b$: two triangles; if $a\geq b$: one triangle.

6. In the SSA ambiguous case with a given obtuse angle $A$ and opposite side $a$, when does a triangle exist?

   Exactly one triangle exists if $a>b$; otherwise (if $a\leq b$) no triangle exists.

7. When solving SSA with the Law of Sines, how do you find the possible second angle from $\sin B$?

   One solution is $B=\sin^{-1}(\ldots)$; the second possible angle is $B'=180^{\circ}-B$. It is valid only if $A+B'<180^{\circ}$.

8. What is the SAS (side-angle-side) area formula for an oblique triangle?

   $$\text{Area}=\frac{1}{2}ab\sin C=\frac{1}{2}bc\sin A=\frac{1}{2}ac\sin B$$

9. Why does the area formula $\frac{1}{2}ab\sin C$ require the SAS configuration?

   Because $C$ must be the angle included between the two known sides $a$ and $b$.

10. What is the Law of Cosines (for side $c$)?

    $$c^{2}=a^{2}+b^{2}-2ab\cos C$$ with analogous forms for $a^{2}$ and $b^{2}$.

11. Which triangle cases are solved using the Law of Cosines?

    The SSS case (all three sides) and the SAS case (two sides and the included angle).

12. How do you solve an SAS triangle?

    Use the Law of Cosines to find the side opposite the given angle, then use the Law of Sines (or Law of Cosines again) for a remaining angle, and subtract from $180^{\circ}$ for the last angle.

13. When solving an SSS triangle, why is it best to find the largest angle first with the Law of Cosines?

    The largest angle is opposite the longest side and may be obtuse; finding it directly with the Law of Cosines avoids the sign ambiguity that the Law of Sines has for obtuse angles.

14. Rearrange the Law of Cosines to solve for angle $C$ given all three sides.

    $$\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}$$

15. What is Heron's formula for the area of a triangle with sides $a,b,c$?

    $$\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is the semiperimeter.

16. How is the semiperimeter $s$ in Heron's formula defined?

    $$s=\frac{a+b+c}{2}$$

17. When is Heron's formula most useful?

    When all three sides (SSS) are known but no angle is given, so you can find the area without first computing an angle.

18. What is the component form of a vector with initial point $P(x_{1},y_{1})$ and terminal point $Q(x_{2},y_{2})$?

    $$\vec{PQ}=\langle x_{2}-x_{1},\, y_{2}-y_{1}\rangle$$

19. What is the magnitude of a vector $\vec{v}=\langle a,b\rangle$?

    $$\|\vec{v}\|=\sqrt{a^{2}+b^{2}}$$

20. What is a position vector?

    A vector whose initial point is the origin; it is represented uniquely by the coordinates of its terminal point $\langle a,b\rangle$.

21. How do you add two vectors $\vec{u}=\langle a,b\rangle$ and $\vec{v}=\langle c,d\rangle$?

    Add componentwise: $\vec{u}+\vec{v}=\langle a+c,\, b+d\rangle$.

22. How do you multiply a vector $\vec{v}=\langle a,b\rangle$ by a scalar $k$?

    $$k\vec{v}=\langle ka,\, kb\rangle$$ which scales the magnitude by $|k|$ and reverses direction if $k<0$.

23. What are the standard unit vectors $\vec{i}$ and $\vec{j}$, and how is $\langle a,b\rangle$ written with them?

    $\vec{i}=\langle 1,0\rangle$, $\vec{j}=\langle 0,1\rangle$; then $\langle a,b\rangle=a\vec{i}+b\vec{j}$.

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