Law of Sines and Cosines, Vectors, and Complex Numbers

The Law of Sines

• An oblique triangle is a triangle that does not contain a right angle.

• The law of sines is applicable when given:

  • Two sides and an angle opposite one of them (SSA)

  • Two angles and any side (AAS or ASA)

Law of Sines Formula

• $\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}$

• Where a, b, c are sides of the triangle and (\alpha, \beta, \gamma) are the opposite angles.

• Three formulas:

  • $\frac{\sin \alpha}{a} = \frac{\sin \beta}{b}$

  • $\frac{\sin \alpha}{a} = \frac{\sin \gamma}{c}$

  • $\frac{\sin \beta}{b} = \frac{\sin \gamma}{c}$

• Alternatively:

  • $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

General Form

In any triangle, the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle to the side opposite that angle.

Problem Solving Rounding Rule

If known sides or angles are stated to a certain accuracy, then unknown sides or angles should be calculated to the same accuracy.

Ambiguous Case (SSA)

When two sides and an angle opposite one of them are given, there might be more than one possible triangle.

• If (\alpha) is acute:

  • If \sin \beta > 1, no triangle exists.

  • If $\sin \beta = 1$, a right triangle is formed.

  • If \sin \beta < 1, two possible choices for the angle (\beta) exist.

• If (\alpha > 90^\circ), a triangle exists if and only if a > b.

Example: Using the law of sines (ASA)

Solve triangle ABC, given $\alpha = 48^\circ$, $\gamma = 57^\circ$, and $b = 47$.

• Find (\beta): $\beta = 180^\circ - 57^\circ - 48^\circ = 75^\circ$

• Find a: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \Rightarrow a = \frac{b \sin \alpha}{\sin \beta} = \frac{47 \sin 48^\circ}{\sin 75^\circ} \approx 36$

• Find c: $\frac{c}{\sin \gamma} = \frac{b}{\sin \beta} \Rightarrow c = \frac{b \sin \gamma}{\sin \beta} = \frac{47 \sin 57^\circ}{\sin 75^\circ} \approx 41$

Example: Using the law of sines (SSA)

Solve triangle ABC, given $\alpha = 67^\circ$, $a = 100$, and $c = 125$.

• Find (\gamma): $\frac{\sin \gamma}{c} = \frac{\sin \alpha}{a} \Rightarrow \sin \gamma = \frac{c \sin \alpha}{a} = \frac{125 \sin 67^\circ}{100} \approx 1.1506$

• Since \sin \gamma > 1, no such triangle can be constructed.

Example: Using the law of sines (SSA) - Two Solutions

Solve triangle ABC, given $a = 12.4$, $b = 8.7$, and $\beta = 36.7^\circ$.

• Find (\alpha): $\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} \Rightarrow \sin \alpha = \frac{a \sin \beta}{b} = \frac{12.4 \sin 36.7^\circ}{8.7} \approx 0.8518$

• Two possible angles for (\alpha): $\alpha<em>1 = \arcsin(0.8518) \approx 58.4^\circ$ and $\alpha</em>2 = 180^\circ - 58.4^\circ = 121.6^\circ$

• For triangle A₁BC: $\gamma<em>1 = 180^\circ - \alpha</em>1 - \beta = 180^\circ - 58.4^\circ - 36.7^\circ = 84.9^\circ$

  • Find c₁: $\frac{c<em>1}{\sin \gamma</em>1} = \frac{a}{\sin \alpha<em>1} \Rightarrow c</em>1 = \frac{a \sin \gamma<em>1}{\sin \alpha</em>1} = \frac{12.4 \sin 84.9^\circ}{\sin 58.4^\circ} \approx 14.5$

• For triangle A₂BC: $\gamma<em>2 = 180^\circ - \alpha</em>2 - \beta = 180^\circ - 121.6^\circ - 36.7^\circ = 21.7^\circ$

  • Find c₂: $\frac{c<em>2}{\sin \gamma</em>2} = \frac{a}{\sin \alpha<em>2} \Rightarrow c</em>2 = \frac{a \sin \gamma<em>2}{\sin \alpha</em>2} = \frac{12.4 \sin 21.7^\circ}{\sin 121.6^\circ} \approx 5.4$

Example: Using an angle of elevation

When the angle of elevation of the sun is 64°, a telephone pole that is tilted at an angle of 9° directly away from the sun casts a shadow 21 feet long on level ground. Approximate the length of the pole.

• Calculate angles: $\beta = 90^\circ - 9^\circ = 81^\circ$ and $\gamma = 180^\circ - 64^\circ - 81^\circ = 35^\circ$

• Use law of sines: $\frac{a}{\sin 64^\circ} = \frac{21}{\sin 35^\circ} \Rightarrow a = \frac{21 \sin 64^\circ}{\sin 35^\circ} \approx 33$

Example: Using bearings

A point P on level ground is 3.0 kilometers due north of a point Q. A runner proceeds in the direction N25°E from Q to a point R, and then from R to P in the direction S70°W. Approximate the distance run.

• Calculate angles: $\angle PQR = 25^\circ$, $\angle PRQ = 70^\circ - 25^\circ = 45^\circ$, and $\angle QPR = 180^\circ - 25^\circ - 45^\circ = 110^\circ$

• Use law of sines to find q: $\frac{q}{\sin 25^\circ} = \frac{3.0}{\sin 45^\circ} \Rightarrow q = \frac{3.0 \sin 25^\circ}{\sin 45^\circ} \approx 1.8$

• Use law of sines to find p: $\frac{p}{\sin 110^\circ} = \frac{3.0}{\sin 45^\circ} \Rightarrow p = \frac{3.0 \sin 110^\circ}{\sin 45^\circ} \approx 4.0$

• Distance run: $p + q \approx 4.0 + 1.8 = 5.8$

The Law of Cosines

• The law of cosines is applicable when given:

  • Two sides and the angle between them (SAS)

  • Three sides (SSS)

Law of Cosines Formula

• $a^2 = b^2 + c^2 - 2bc \cos \alpha$

• $b^2 = a^2 + c^2 - 2ac \cos \beta$

• $c^2 = a^2 + b^2 - 2ab \cos \gamma$

General Form

The square of the length of any side of a triangle equals the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle between them.

Pythagorean Theorem

If (\alpha = 90^\circ), then the law of cosines reduces to the Pythagorean theorem: $a^2 = b^2 + c^2$

Problem Solving Suggestions

• Given two sides and the included angle, use the law of cosines to find the third side, then use the law of sines to find another angle (find the angle opposite the shortest side first).

• Given three sides, find the largest angle first (opposite the longest side).

Example: Using the law of cosines (SAS)

Solve triangle ABC, given $a = 5.0$, $c = 8.0$, and $\beta = 77^\circ$.

• Find b: $b^2 = a^2 + c^2 - 2ac \cos \beta = (5.0)^2 + (8.0)^2 - 2(5.0)(8.0) \cos 77^\circ \approx 71.0 \Rightarrow b \approx \sqrt{71.0} \approx 8.4$

• Find (\alpha): $\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} \Rightarrow \sin \alpha = \frac{a \sin \beta}{b} = \frac{5.0 \sin 77^\circ}{\sqrt{71.0}} \approx 0.5782 \Rightarrow \alpha \approx \arcsin(0.5782) \approx 35.3^\circ \approx 35^\circ$

• Find (\gamma): $\gamma = 180^\circ - \alpha - \beta \approx 180^\circ - 35^\circ - 77^\circ = 68^\circ$

Example: Using the law of cosines (SSS)

If triangle ABC has sides $a = 90$, $b = 70$, and $c = 40$, approximate angles (\alpha, \beta), and (\gamma).

• Find (\alpha): $\cos \alpha = \frac{b^2 + c^2 - a^2}{2bc} = \frac{70^2 + 40^2 - 90^2}{2(70)(40)} = -\frac{2}{7} \Rightarrow \alpha = \arccos(-\frac{2}{7}) \approx 106.6^\circ \approx 107^\circ$

• Find (\beta): $\cos \beta = \frac{a^2 + c^2 - b^2}{2ac} = \frac{90^2 + 40^2 - 70^2}{2(90)(40)} = \frac{3}{7} \Rightarrow \beta = \arccos(\frac{3}{7}) \approx 48.2^\circ \approx 48^\circ$

• Find (\gamma): $\cos \gamma = \frac{a^2 + b^2 - c^2}{2ab} = \frac{90^2 + 70^2 - 40^2}{2(90)(70)} = \frac{11}{14} \Rightarrow \gamma = \arccos(\frac{11}{14}) \approx 38.2^\circ \approx 25^\circ$

Example: Approximating the diagonals of a parallelogram

A parallelogram has sides of lengths 30 centimeters and 70 centimeters and one angle of measure 65°. Approximate the length of each diagonal.

• Diagonal AC: $(AC)^2 = 30^2 + 70^2 - 2(30)(70) \cos 65^\circ \approx 4025 \Rightarrow AC \approx \sqrt{4025} \approx 63$

• Diagonal BD: $(BD)^2 = 30^2 + 70^2 - 2(30)(70) \cos 115^\circ \approx 7575 \Rightarrow BD \approx \sqrt{7575} \approx 87$

Example: Finding the length of a cable

A vertical pole 40 feet tall stands on a hillside that makes an angle of 17° with the horizontal. Approximate the minimal length of cable that will reach from the top of the pole to a point 72 feet downhill from the base of the pole.

• $\angle ABD = 90^\circ - 17^\circ = 73^\circ$ and $\angle ABC = 180^\circ - 73^\circ = 107^\circ$

• $(AC)^2 = 72^2 + 40^2 - 2(72)(40) \cos 107^\circ \approx 8468 \Rightarrow AC \approx \sqrt{8468} \approx 92$

Area of a Triangle

• The area of a triangle equals one-half the product of the lengths of any two sides and the sine of the angle between them.

• $\Delta = \frac{1}{2}bc \sin \alpha = \frac{1}{2}ac \sin \beta = \frac{1}{2}ab \sin \gamma$

Heron's Formula

• $\Delta = \sqrt{s(s - a)(s - b)(s - c)}$

• Where s is one-half the perimeter, i.e. $s = \frac{1}{2}(a + b + c)$

Vectors

• A vector is a quantity that has both magnitude and direction.

• Equal vectors have the same magnitude and direction.

• A scalar is a quantity that has only magnitude.

• Velocity and force are examples of vectors.

Vector Addition

• Triangle Law: add vectors by placing the initial point of the second vector on the terminal point of the first.

• Parallelogram Law: If PQ and PR are two forces acting at P, then PS is the resultant force, where RPQS is parallelogram.

Scalar Multiplication

If m is a scalar and v is a vector, then mv is a vector whose magnitude is |m| times ||v|| and whose direction is either the same as that of v (if m > 0) or opposite that of v (if m < 0).

Component Form of Vectors

• A vector in the xy-plane can be represented by an ordered pair of real numbers (a₁, a₂).

• The magnitude of the vector a = (a₁, a₂) is $|a| = \sqrt{a<em>1^2 + a</em>2^2}$.

• Addition: $(a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂)$

• Scalar Multiple: $m(a₁, a₂) = (ma₁, ma₂)$

• The zero vector is 0 = (0, 0).

• The negative of a vector a = (a₁, a₂) is -a = (-a₁, -a₂).

• Vector subtraction is defined by a - b = a + (-b).

Vector Properties

• a + b = b + a

• a + (b + c) = (a + b) + c

• a + 0 = a

• a + (-a) = 0

• m(a + b) = ma + mb

• (m + n)a = ma + na

• (mn)a = m(na) = n(ma)

• 1a = a

• 0a = 0 = m0

Unit Vectors i and j

• i = (1, 0) and j = (0, 1)

• a = (a₁, a₂) = a₁i + a₂j

• a₁ is the horizontal component and a₂ is the vertical component of the vector a.

• (a₁i + a₂j) + (b₁i + b₂j) = (a₁ + b₁)i + (a₂ + b₂)j

• (a₁i + a₂j) - (b₁i + b₂j) = (a₁ - b₁)i + (a₂ - b₂)j

• m(a₁i + a₂j) = (ma₁)i + (ma₂)j

Components Using Magnitude and Direction

• $a_1 = |a| \cos \theta$

• $a_2 = |a| \sin \theta$

• where (\theta) is the angle in standard position from the positive x-axis to the vector a = (a₁, a₂).

• Therefore, $a = |a|(\cos \theta i + \sin \theta j)$

Complex Numbers

Represent complex numbers geometrically by using points in a coordinate plane, called a complex plane. The x-axis is the real axis, and the y-axis is the imaginary axis.

Absolute Value of a Complex Number

• If z = a + bi, then $|a + bi| = \sqrt{a^2 + b^2}$.

Trigonometric (Polar) Form

• $z = r(\cos \theta + i \sin \theta) = r \text{cis} \theta$

• Where $r = |z| = \sqrt{a^2 + b^2}$ and (\theta) is an argument of z.

• Exponential Form: $z = re^{i\theta}$.

Theorem on Products and Quotients

If $z<em>1 = r</em>1(\cos \theta<em>1 + i \sin \theta</em>1)$ and $z<em>2 = r</em>2(\cos \theta<em>2 + i \sin \theta</em>2)$, then:

• $z<em>1 z</em>2 = r<em>1 r</em>2 [\cos(\theta<em>1 + \theta</em>2) + i \sin(\theta<em>1 + \theta</em>2)]$

• $\frac{z<em>1}{z</em>2} = \frac{r<em>1}{r</em>2} [\cos(\theta<em>1 - \theta</em>2) + i \sin(\theta<em>1 - \theta</em>2)], z_2 \neq 0$

De Moivre's Theorem

For every integer n:

$[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)$

Theorem on nth Roots

If $z = r(\cos \theta + i \sin \theta)$, then z has exactly n distinct nth roots $w<em>0, w</em>1, w<em>2, …, w</em>{n-1}$, where:

$w_k = \sqrt[n]{r} \left[ \cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right) \right]$

Or, in degrees:

$w_k = \sqrt[n]{r} \left[ \cos \left( \frac{\theta + 360^\circ k}{n} \right) + i \sin \left( \frac{\theta + 360^\circ k}{n} \right) \right]$

where k = 0, 1, …, n - 1.

The Law of Sines

• Used for oblique triangles (no right angle).

• Applicable when given:

  • Two sides and an opposite angle (SSA)

  • Two angles and any side (AAS or ASA)

• Formula: $\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}$

• Ambiguous Case (SSA): 0, 1, or 2 possible triangles.

The Law of Cosines

• Applicable when given:

  • Two sides and the included angle (SAS)

  • Three sides (SSS)

• Formulas:

  • $a^2 = b^2 + c^2 - 2bc \cos \alpha$

  • $b^2 = a^2 + c^2 - 2ac \cos \beta$

  • $c^2 = a^2 + b^2 - 2ab \cos \gamma$

Area of a Triangle

• Formula: $\Delta = \frac{1}{2}bc \sin \alpha = \frac{1}{2}ac \sin \beta = \frac{1}{2}ab \sin \gamma$

• Heron's Formula: $\Delta = \sqrt{s(s - a)(s - b)(s - c)}$ where $s = \frac{1}{2}(a + b + c)$

Vectors

• Quantity with magnitude and direction.

• Vector Addition: Triangle Law, Parallelogram Law.

• Scalar Multiplication: Changes magnitude, may reverse direction.

• Component Form: a = (a₁, a₂), $|a| = \sqrt{a<em>1^2 + a</em>2^2}$

• Unit Vectors: i = (1, 0), j = (0, 1); a = a₁i + a₂j

• Components: $a<em>1 = |a| \cos \theta$, $a</em>2 = |a| \sin \theta$

Complex Numbers

• Represented on a complex plane (real and imaginary axes).

• Absolute Value: $|a + bi| = \sqrt{a^2 + b^2}$

• Polar Form: $z = r(\cos \theta + i \sin \theta) = r \text{cis} \theta$, where $r = |z| = \sqrt{a^2 + b^2}$

• Products and Quotients:

  • $z<em>1 z</em>2 = r<em>1 r</em>2 [\cos(\theta<em>1 + \theta</em>2) + i \sin(\theta<em>1 + \theta</em>2)]$

  • $\frac{z<em>1}{z</em>2} = \frac{r<em>1}{r</em>2} [\cos(\theta<em>1 - \theta</em>2) + i \sin(\theta<em>1 - \theta</em>2)], z_2 \neq 0$

• De Moivre's Theorem: $[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)$

• nth Roots: w_k = \sqrt[n]{r} \left[ \cos \left( \frac{\theta + 360^\circ k}{n} \right) + i \sin \left( \frac{\

The Law of Sines

• Used for oblique triangles (no right angle).

• Applicable when given:

  • Two sides and an opposite angle (SSA)

  • Two angles and any side (AAS or ASA)

• Formula: \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}$</p></li><li><p>Ambiguous Case (SSA): 0, 1, or 2 possible triangles.</p></li></ul><h5 id="fe4c9880-a984-4941-b38a-b90da8c50444" data-toc-id="fe4c9880-a984-4941-b38a-b90da8c50444" collapsed="false" seolevelmigrated="true">The Law of Cosines</h5><ul><li><p>Applicable when given:</p><ul><li><p>Two sides and the included angle (SAS)</p></li><li><p>Three sides (SSS)</p></li></ul></li><li><p>Formulas:</p><ul><li><p>$a^2 = b^2 + c^2 - 2bc \cos \alpha$</p></li><li><p>$b^2 = a^2 + c^2 - 2ac \cos \beta$</p></li><li><p>$c^2 = a^2 + b^2 - 2ab \cos \gamma$</p></li></ul></li></ul><h5 id="d8a7afbe-5841-47c0-a9b6-57a7e7bb5560" data-toc-id="d8a7afbe-5841-47c0-a9b6-57a7e7bb5560" collapsed="false" seolevelmigrated="true">Area of a Triangle</h5><ul><li><p>Formula:$\Delta = \frac{1}{2}bc \sin \alpha = \frac{1}{2}ac \sin \beta = \frac{1}{2}ab \sin \gamma$</p></li><li><p>Heron's Formula:$\Delta = \sqrt{s(s - a)(s - b)(s - c)}$where$s = \frac{1}{2}(a + b + c)$</p></li></ul><h5 id="8981f244-747a-48d1-abae-078705213b0c" data-toc-id="8981f244-747a-48d1-abae-078705213b0c" collapsed="false" seolevelmigrated="true">Vectors</h5><ul><li><p>Quantity with magnitude and direction.</p></li><li><p>Vector Addition: Triangle Law, Parallelogram Law.</p></li><li><p>Scalar Multiplication: Changes magnitude, may reverse direction.</p></li><li><p>Component Form: a = (a₁, a₂),$|a| = \sqrt{a1^2 + a2^2}$</p></li><li><p>Unit Vectors: i = (1, 0), j = (0, 1); a = a₁i + a₂j</p></li><li><p>Components:$a1 = |a| \cos \theta$,$a2 = |a| \sin \theta$</p></li></ul><h5 id="46a723ba-0b91-4087-8023-b3a2538b4cb6" data-toc-id="46a723ba-0b91-4087-8023-b3a2538b4cb6" collapsed="false" seolevelmigrated="true">Complex Numbers</h5><ul><li><p>Represented on a complex plane (real and imaginary axes).</p></li><li><p>Absolute Value:$|a + bi| = \sqrt{a^2 + b^2}$</p></li><li><p>Polar Form:$z = r(\cos \theta + i \sin \theta) = r \text{cis} \theta$, where$r = |z| = \sqrt{a^2 + b^2}$</p></li><li><p>Products and Quotients:</p><ul><li><p>$z1 z2 = r1 r2 [\cos(\theta1 + \theta2) + i \sin(\theta1 + \theta2)]$</p></li><li><p>$\frac{z1}{z2} = \frac{r1}{r2} [\cos(\theta1 - \theta2) + i \sin(\theta1 - \theta2)], z_2 \neq 0$</p></li></ul></li><li><p>De Moivre's Theorem:$[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)$</p></li><li><p>nth Roots:$w_k = \sqrt[n]{r} \left[ \cos \left( \frac{\theta + 360^\circ k}{n} \right) + i \sin \left( \frac{\