Chapter 10 — Further Applications of Trigonometry

For any triangle with sides $a, b, c$ opposite angles $A, B, C$:

Use when you know:

• AAS (two angles + non-included side)
• ASA (two angles + included side)
• SSA (two sides + non-included angle) — ambiguous case

(Half the product of two sides times the sine of the included angle.)

\(a^2 = b^2 + c^2 - 2bc\cos A\) \(b^2 = a^2 + c^2 - 2ac\cos B\) \(c^2 = a^2 + b^2 - 2ab\cos C\)

Use when you know:

• SAS (two sides + included angle)
• SSS (three sides)

Note: When $C = 90°$, this reduces to the Pythagorean Theorem.

For a triangle with sides $a, b, c$, let the semi-perimeter be:

Then the area is:

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

A point in polar coordinates is written $(r, \theta)$ where:

• $r$ = directed distance from the origin (pole)
• $\theta$ = angle from the positive $x$-axis (polar axis)

If $r < 0$, the point is plotted in the opposite direction.

Unlike Cartesian, polar representations are not unique:

$(r, \theta) = (r, \theta + 2n\pi) = (-r, \theta + \pi + 2n\pi)$

Polar → Rectangular:

Rectangular → Polar:

(Choose $\theta$ in the correct quadrant.)

Converting equations:

| Rectangular | Polar | |————-|——-| | $x^2 + y^2 = a^2$ | $r = a$ (circle) | | $y = x$ | $\theta = \pi/4$ (line) | | $x = a$ | $r\cos\theta = a$ | | $y = a$ | $r\sin\theta = a$ | | $x^2 + y^2 = ax$ | $r = a\cos\theta$ (circle through origin) | | $x^2 + y^2 = ay$ | $r = a\sin\theta$ (circle through origin) |

| Equation | Shape | Notes | |———-|——-|——-| | $r = a$ | Circle centered at origin | Radius $a$ | | $\theta = \alpha$ | Line through origin | Angle $\alpha$ | | $r = a\cos\theta$ | Circle | Center $(a/2, 0)$, diameter $a$ | | $r = a\sin\theta$ | Circle | Center $(0, a/2)$, diameter $a$ | | $r = a + b\cos\theta$ | Limaçon | $a/b < 1$: inner loop; $= 1$: cardioid; $> 1$: dimpled or convex | | $r = a + a\cos\theta$ | Cardioid | Heart-shaped, passes through origin | | $r = a\cos(n\theta)$ | Rose | $n$ petals if $n$ odd; $2n$ petals if $n$ even | | $r^2 = a^2\cos(2\theta)$ | Lemniscate | Figure-eight, centered at origin | | $r = a\theta$ | Archimedean spiral | Distance increases linearly with angle |

A complex number $z = a + bi$ is plotted in the complex plane with $a$ on the horizontal (real) axis and $b$ on the vertical (imaginary) axis.

Modulus (absolute value):

\(|z| = \sqrt{a^2 + b^2}\)

where $r = |z|$ and $\theta = \arg(z) = \tan^{-1}(b/a)$ (adjusted for quadrant).

If $z_1 = r_1\,\text{cis}\,\theta_1$ and $z_2 = r_2\,\text{cis}\,\theta_2$:

Multiply: multiply moduli, add arguments.
Divide: divide moduli, subtract arguments.

for any positive integer $n$.

The $n$ distinct $n$th roots of $z = r\,\text{cis}\,\theta$ are:

The roots are equally spaced around a circle of radius $r^{1/n}$, separated by $\dfrac{2\pi}{n}$ radians.

A parametric curve is defined by:

where $t$ is a parameter (often representing time). As $t$ varies, the point $(x, y)$ traces out a curve in the plane.

Circle: $x = h + r\cos t$, $y = k + r\sin t$

Ellipse: $x = h + a\cos t$, $y = k + b\sin t$

Projectile Motion (launched at angle $\alpha$ with speed $v_0$ from height $h$):

\(x = (v_0\cos\alpha)t\) \(y = -\frac{1}{2}gt^2 + (v_0\sin\alpha)t + h\)

where $g \approx 9.8\,\text{m/s}^2$ (or $32\,\text{ft/s}^2$).

A vector has both magnitude (length) and direction.

Notation: $\mathbf{v}$, $\vec{v}$, or $\langle a, b \rangle$.

A scalar has magnitude only (a real number).

Two vectors are equal if they have the same magnitude and direction, regardless of position.

The vector from $P(x_1, y_1)$ to $Q(x_2, y_2)$:

Magnitude:

Direction angle $\theta$:

\(\tan\theta = \frac{b}{a}, \quad \mathbf{v} = \|\mathbf{v}\|\langle\cos\theta, \sin\theta\rangle\)

Let $\mathbf{u} = \langle u_1, u_2 \rangle$ and $\mathbf{v} = \langle v_1, v_2 \rangle$, and let $c$ be a scalar.

Geometric interpretation:

• Addition: Tip-to-tail method or parallelogram rule.

• Scalar multiplication: Stretches (if $

  c

  > 1$) or shrinks (if $

  c

  < 1$) the vector; reverses direction if $c < 0$.

A unit vector has magnitude 1. The unit vector in the direction of $\mathbf{v}$ is:

Standard unit vectors:

Any vector: $\mathbf{v} = \langle a, b \rangle = a\mathbf{i} + b\mathbf{j}$

From magnitude and direction:

\(\mathbf{v} = \|\mathbf{v}\|\cos\theta\,\mathbf{i} + \|\mathbf{v}\|\sin\theta\,\mathbf{j}\)

Algebraic definition:

Geometric definition:

where $\theta$ is the angle between the vectors.

Angle between vectors:

\(\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\)

Key properties:

• Orthogonal (perpendicular): $\mathbf{u} \cdot \mathbf{v} = 0$
• Commutative: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$
• Distributive: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$
• $\mathbf{v} \cdot \mathbf{v} = |\mathbf{v}|^2$

Projection of $\mathbf{u}$ onto $\mathbf{v}$:

Scalar component:

\(\text{comp}_{\mathbf{v}}\,\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}\)