Trigonometry Handbook Notes (Version 2.4)

Chapter 1 – Functions and Special Angles

• What is Trigonometry?

  • The study of measures of triangles: sides, angles, and their relationships.

  • Modern approach connects right triangles with circles, especially the Unit Circle (radius 1).

• Angles: radians and degrees

  • 1 radian is the angle subtended by an arc whose length equals the radius.

  • Arc length formula (in radians): $s = r\theta$ where $\theta$ is in radians.

• Angle definitions (on Cartesian plane)

  • Initial side, terminal side, vertex, standard position (vertex at origin and initial side on the positive x-axis).

  • Polar axis = positive x-axis; polar angle = angle from polar axis to terminal side; reference angle = acute angle with the axis; coterminal angles differ by multiples of $360^\circ$ or $2\pi$; quadrantal angle ends on an axis.

• Trigonometric functions (on the x- and y-axes)

  • For a point on the unit circle, $x = \cos\theta$, $y = \sin\theta$.

  • Reciprocal relations: $\sin\theta = \frac{1}{\csc\theta},\quad \cos\theta = \frac{1}{\sec\theta},\quad \tan\theta = \frac{1}{\cot\theta}.$

  • Tangent as sine over cosine: $\tan\theta = \frac{\sin\theta}{\cos\theta}.$

• Sine-Cosine relationship (identities)

  • $\sin\theta = \cos\left(\frac{\pi}{2}-\theta\right),\quad \cos\theta = \sin\left(\frac{\pi}{2}-\theta\right),\quad \tan\theta = \cot\left(\frac{\pi}{2}-\theta\right).$

• Key angles in radians and degrees

  • Common values (Unit Circle):

  • $\sin 0^\circ = 0,\quad \cos 0^\circ = 1,\quad \tan 0^\circ = 0.$

  • $\sin 30^\circ = \frac{1}{2},\quad \cos 30^\circ = \frac{\sqrt{3}}{2},\quad \tan 30^\circ = \frac{1}{\sqrt{3}}.$

  • $\sin 45^\circ = \frac{\sqrt{2}}{2},\quad \cos 45^\circ = \frac{\sqrt{2}}{2},\quad \tan 45^\circ = 1.$

  • $\sin 60^\circ = \frac{\sqrt{3}}{2},\quad \cos 60^\circ = \frac{1}{2},\quad \tan 60^\circ = \sqrt{3}.$

  • $\sin 90^\circ = 1,\quad \cos 90^\circ = 0,\quad \tan 90^\circ \text{ undefined}.$

• Cofunctions(functions whose values are equal for complementary angles)

  • $\sin\theta$ and $\cos\theta$ are related by cofunctions (e.g., $\sin\theta = \cos(\tfrac{\pi}{2}-\theta)$).

• Unit Circle

  • A diagram where, for any angle, the point on the circle of radius 1 has coordinates $(\cos\theta, \sin\theta)$.

  • Sine and cosine values recur at common angles; helps identify signs in other quadrants.

• Functions in a Right Triangle (SOH-CAH-TOA)

  • Sine/$\sin\theta$ = $\frac{opposite}{hypotenuse}$ =

  • Cosine/$\cos\theta$ = $\frac{adjacent}{hypotenuse}$ =

  • Tangent/$\tan\theta$ = $\frac{opposite}{adjacent}$ =

• Trigonometric functions of special angles (table)

  • For angles $\theta$ at $0,\;\tfrac{\pi}{6},\;\tfrac{\pi}{4},\;\tfrac{\pi}{3},\;\tfrac{\pi}{2}$, values are given as multiples of radicals (e.g., $\sqrt{2}/2$, $\sqrt{3}/2$).

• Quadrants II, III, IV (signs)

  • Quadrant II: sin positive, cos negative, tan negative.

  • Quadrant III: sin negative, cos negative, tan positive.

  • Quadrant IV: sin negative, cos positive, tan negative.

• Problems in Quadrants II–IV

  • Approach: draw the angle, find reference angle, compute the value from the corresponding angle in Quadrant I, then assign the sign from the quadrant.

• Angles of Depression and Inclination

  • Depression: angle below the horizontal; Inclination: angle above the horizontal.

  • Examples illustrate using tangent and complement relationships to find angles.

Chapter 2 – Graphs of Trig Functions

• Key graph characteristics (parent vs. transformed)

  • General form: $f(x) = A\sin(Bx - C) + D,\quad A, B, C, D\in\mathbb{R}$ where $A$ is amplitude (absolute value), $B$ affects period, $C$ phase shift, $D$ vertical shift.

  • Amplitude: $|A|$. Applies to sine and cosine; other functions do not have amplitude in the same sense.

  • Period: for sine, cosine, secant, cosecant, period is $\frac{2\pi}{|B|}.$ for tangent and cotangent, period is $\frac{\pi}{|B|}.$ In general, $\text{Period} = \frac{2\pi}{|B|}.$ (For the general form, replace as needed for the specific function.)

  • Frequency (relevant to sine/cosine): $\text{Frequency} = \frac{|B|}{2\pi}.$ 1 Hz = 1 cycle/second.

  • Phase shift: horizontal shift; for $f(x)=A\sin(Bx-C)+D,$ set $Bx-C=0$ to find the shift $x = \dfrac{C}{B}$; sign convention: minus in the argument yields a shift to the right;
    plus yields a shift to the left.

  • Vertical shift: $D$; shift of the midline from $y=0$ to $y=D$.

• Graphing basic trig functions

  • Sine and cosine are periodic with amplitude and a midline; tangent and cotangent have vertical asymptotes; secant and cosecant have asymptotes at zeros of cosine and sine respectively.

  • Intersections of function with its reciprocal occur at extrema (e.g., sine and cosecant intersect at maxima/minima when sine is ±1).

• Summary of characteristics (per function)

  • Sine: domain $(-\infty,\infty)$, range $[-1,1]$, period $2\pi$, odd function.

  • Cosine: domain $(-\infty,\infty)$, range $[-1,1]$, period $2\pi$, even function.

  • Tangent: domain excludes odd multiples of $\tfrac{\pi}{2}$, period $\pi$, odd function.

  • Cotangent: domain excludes multiples of $\pi$, period $\pi$, odd function.

  • Secant and cosecant: period $2\pi$ with vertical asymptotes where cosine or sine is zero.

• Graphing a general sine or cosine (examples)

  • General form: $y = A\sin(Bx - C) + D\quad \text{or}\quad y = A\cos(Bx - C) + D.$

  • Determine amplitude $|A|$, period $\dfrac{2\pi}{|B|}$, phase shift $\dfrac{C}{B}$, vertical shift $D$.

  • Example outline: identify midline $y=D$, one cycle length (period) to place key points, then sketch the wave.

• Graphs and features

  • One sine wave without vertical shift: step-by-step construction using phase shift, then period, then key points, then reflection/translation as needed.

  • Sine and cosine graphs: distinction in positions of zero crossings and extrema; tangent/cotangent steps revolve around asymptotes and zeros.

Chapter 3 – Inverse Trigonometric Functions

• What are inverse trig functions?

  • Inverse trig functions are denoted with a superscript -1 or with an arc prefix, e.g., $\sin^{-1}$ or $\arcsin$.

  • They ask: which angle $\theta$ has a given function value? E.g., $\sin^{-1}(0.5)$ asks for the angle with sine 0.5.

• Principal values and ranges

  • Inverse trig functions are multi-valued over all reals due to periodicity; principal values restrict to specific quadrants:

  • $\sin^{-1}$: principal values in $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.

  • $\cos^{-1}$: principal values in $[0, \pi]$.

  • $\tan^{-1}$: principal values in $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$.

• Graphs and principal values (overview)

  • Graphs of inverse trig functions show their principal values; outputs are within the restricted ranges above.

• Problems involving inverse trig functions

  • Example: Find the principal value of $\tan^{-1}(\tan 23\circ)$: consider quadrant where tan inverse is defined (Q1 or Q4 for principal value), adjust for periodicity.

  • Example: Find $\sin^{-1}(\cos \theta)$ by using the identity $\sin \alpha = \cos(\tfrac{\pi}{2}-\alpha)$ or by comparing quadrants and using Pythagorean relations.

• Notes on compositions with inverse trig functions

  • When inverse trig is inside a composition, draw the triangle defined by the inner function, then evaluate the outer function.

Chapter 4 – Key Angle Formulas

• Angle addition formulas

  • $\sin(A+B) = \sin A\cos B + \cos A\sin B$

  • $\cos(A+B) = \cos A\cos B - \sin A\sin B$

  • $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}$

• Double angle formulas

  • $\sin(2A) = 2\sin A\cos A$

  • $\cos(2A) = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

  • $\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}$

• Half angle formulas

  • $\sin\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{2}},\quad \cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 + \cos A}{2}}$

  • $\tan\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$

• Other formulas

  • Power-reducing, product-to-sum, and sum-to-product formulas (listed in Appendix A); signs depend on quadrant.

• Examples (illustrative, not all details)

  • Example 4.1: Use product-to-sum or sum-to-product to evaluate expressions like $\cos 175^\circ \cos 55^\circ$.

  • Example 4.7: Use half-angle to evaluate $\cos(5\pi/12)$, noting quadrant for sign.

  • Examples 4.3–4.8 demonstrate angle-changes, quadrant signs, and exact-value evaluations.

Chapter 5 – Identities and Equations

• Verifying identities

  • Goal: manipulate one side (Side A) to look like the other side (Side B) using algebraic/trigonometric rules.

  • Steps:

  • Decide which side to transform (prefer a side with multiple angles, or with more complex structure).

  • Use Pythagorean identities to replace $\sin$ with $\cos$, or vice versa;

  • Convert multi-angle or half-angle expressions to single-angle forms; look for 1’s to factor.

  • Rewrite both sides in terms of sines/cosines and factor where possible; compare.

• Techniques (examples)

  • Investigate one or both sides to gain insight; transform one side into the other.

  • Break a fraction into pieces to simplify the numerator/denominator.

  • Get a common denominator on one side and simplify to match the other side.

• Solving trigonometric equations

  • Steps: arrange the equation, factor as needed, use replacement (Pythagorean, etc.), check for extraneous solutions.

  • Watch for extraneous solutions caused by squaring or restricting domains; test in original equation.

  • Infinite solutions: add multiples of the period (e.g., $2\pi k$).

  • Solutions in an interval: number of solutions depends on coefficient and period; example considerations for $2\pi k$ periodicity.

Chapter 6 – Solving an Oblique Triangle

• Laws of Sines and Cosines

  • Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$

  • Law of Cosines: $a^2 = b^2 + c^2 - 2bc\cos A,<br>\quad b^2 = a^2 + c^2 - 2ac\cos B,<br>\quad c^2 = a^2 + b^2 - 2ab\cos C.$

  • Interpretation: relates sides and opposite angles; helps solve oblique (non-right) triangles.

• Solving strategies (SSS, SAS, ASA/AAS, SSA)

  • SSS: compute an angle via Law of Cosines, then remaining angles with Law of Sines.

  • SAS: compute the third side with Law of Cosines, then a remaining angle with Law of Sines.

  • ASA/AAS: sum of known angles gives the third angle; use Law of Sines for the remaining sides.

  • SSA (ambiguous case): two possible triangles; approach with Height Comparison, Butterfly Method, or Sine Validity (see Appendix B).

• Height (ambiguous case) method

  • Compute height $h = b\sin A$ and compare to side $a$; determine cases (no triangle, one right triangle, two triangles).

  • If two triangles possible, compute two cases for angle $B$ via $\sin B$ and proceed to compute other angles and sides.

• Butterfly method

  • A quadratic in the unknown side arises from the Law of Cosines; solve for possible $c$ values, then use Law of Sines to find angles.

• Sine validity method

  • Use the Law of Sines to compute potential angle $B$ and check whether sine value is valid; determines number of triangles.

• Bearings (directional angles)

  • Bearings measured clockwise or counterclockwise from north or south; convert to reference angles and polar angles to solve.

• Bearings – Examples

  • Example problems include distance calculations using bearings and the Law of Cosines or Sines.

Chapter 7 – Area of a Triangle

• Geometry formulas

  • Geometry formula: area = (1/2) × base × height, where base can be any side and height is the corresponding altitude.

  • Heron’s formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter.

• Trigonometric area formulas (derived from geometry formula)

  • Using two sides and the included angle: $\text{Area} = \tfrac{1}{2} ab \sin C,$ and cyclic variants.

  • Using two angles and a side: forms depend on the known information; examples illustrate substitution using the Law of Sines.

• Coordinate geometry method

  • Given vertices $(x1,y1),(x2,y2),(x3,y3)$, area is:
    $\text{Area} = \tfrac{1}{2}\,|x<em>1(y</em>2 - y<em>3) + x</em>2(y<em>3 - y</em>1) + x<em>3(y</em>1 - y_2)|.$

• Examples

  • Heron’s formula example computations; coordinate geometry area example; using multiple methods to verify consistency.

Chapter 8 – Polar Coordinates

• Basics

  • Point description by $(r,\theta)$: $r$ is distance from origin; $\theta$ is the polar angle; $\theta$ typically in radians.

  • Rectangular-to-polar conversions: $x = r\cos\theta,\quad y = r\sin\theta,\quad r^2 = x^2+y^2,\quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$ (adjust quadrant).

• Polar form of complex numbers

  • Complex number $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$ (Euler’s formula).

  • Operations on complex numbers in polar form: multiply magnitudes and add angles; division divides magnitudes and subtracts angles; powers multiply angles; roots use De Moivre’s Theorem.

• De Moivre’s Theorem

  • For $z = r(\cos\theta + i\sin\theta)$, $z^n = r^n(\cos(n\theta) + i\sin(n\theta)).$

• Roots of unity (polar roots)

  • The $n$-th roots of unity lie on the circle with radius $1$ (or magnitude appropriate) and angles spaced by $\frac{2\pi}{n}$.

• Examples (illustrative)

  • Compute $3^3$ in polar form; compute $\sqrt{5} \ e^{i\theta}$ forms; use De Moivre’s for powers and roots.

Chapter 9 – Polar Functions

• Polar graph components

  • Parts of a polar graph include the pole (origin), polar axis, and graph of $r$ against $\theta$.

• Symmetry tests (in polar graphing)

  • Symmetry about the polar axis, the line $\theta = \frac{\pi}{2}$ (the line perpendicular to the polar axis), or through the pole; tests involve replacing $\theta$ with $-\theta$, $\pi - \theta$, etc., and checking for equivalence.

• Graphing methods

  • Point plotting: compute $r$ for various $\theta$ and plot points $(r,\theta)$; identify curves (circles, roses, limaçons, lemniscates, spirals).

  • Calculator method: set mode to polar, input equation in $r = f(\theta)$, plot with appropriate window and step.

• Graph types

  • Circle, Rose (roses have $\,r = a\sin(n\theta)\,$ or $r = a\cos(n\theta)$; number of petals depends on $n$ being even/odd), Limaçon (with/without inner loop), Lemniscate (Bernoulli), Spirals (Archimedean, Fermat’s, Hyperbolic, Lituus).

• Converting between polar and rectangular forms

  • Rectangular to polar: $x = r\cos\theta,$ $y = r\sin\theta$; $r = \sqrt{x^2+y^2}$, $\theta = \tan^{-1}(y/x)$ (quadrant-corrected).

  • Polar to rectangular: $x = r\cos\theta$, $y = r\sin\theta$.

• Complex number perspective in polar graphs (brief)

  • Polar form $re^{i\theta}$; relationships with rectangular forms via Euler’s formula.

• Examples

  • Rose, Cardioid, Limaçon shapes; using symmetry and petal counts to sketch.

• Parametric and other connections

  • Parametric curves can be described in polar form; examples include classical curves such as cycloids, Deltoids, Nephroids, Astroids.

Chapter 10 – Vectors

• What is a vector?

  • A quantity with magnitude and direction (e.g., wind, force).

• Special unit vectors

  • Unit vectors in directions of coordinate axes (i.e., i, j, k) with magnitude 1.

• Vector components and magnitude

  • If a vector has endpoints $(x1,y1,z1)$ and $(x2,y2,z2)$, displacement is $\langle x2-x1, y2-y1, z2-z1\rangle$.

  • Magnitude: $|\mathbf{v}| = \sqrt{(v<em>x)^2+(v</em>y)^2+(v_z)^2}.$

• Vector notation and basic properties

  • Vector components in 2D: $\mathbf{u}=\langle u<em>x,u</em>y\rangle,\; |\mathbf{u}| = \sqrt{u<em>x^2+u</em>y^2}.$

  • Unit vector in direction of $\mathbf{v}$: $\hat{v}=\frac{\mathbf{v}}{|\mathbf{v}|}.$

• Vector addition, subtraction, and scalar multiplication

  • Vector addition: component-wise addition; $\mathbf{u}+\mathbf{v}$.

  • Dot product: $\mathbf{u}\cdot \mathbf{v} = u<em>x v</em>x + u<em>y v</em>y (+ u<em>z v</em>z).$

  • Cross product (3D): $\mathbf{u}\times\mathbf{v} = \begin{vmatrix} \mathbf{i} &amp; \mathbf{j} &amp; \mathbf{k} \ u<em>x &amp; u</em>y &amp; u<em>z \ v</em>x &amp; v<em>y &amp; v</em>z \end{vmatrix},$ which yields a vector perpendicular to both; magnitude: $|\mathbf{u}\times\mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin\theta.$

• Dot product properties

  • Zero property: $\mathbf{u}\cdot\mathbf{v}=0$ iff $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.

  • Distributive: $\mathbf{u}\cdot(\mathbf{v}+\mathbf{w}) = \mathbf{u}\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{w}$.

  • Magnitude square: $\mathbf{u}\cdot\mathbf{u} = |\mathbf{u}|^2$.

• Vector projection and components

  • Projection of $\mathbf{u}$ onto $\mathbf{v}$: $\text{proj}_{\mathbf{v}}\mathbf{u} = \left(\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\right) \mathbf{v}.$

  • Orthogonal components: any vector $\mathbf{u}$ can be decomposed as $\mathbf{u}=\text{proj}{\mathbf{v}}\mathbf{u} + \mathbf{u}{\perp}$ where $\mathbf{u}_{\perp}$ is orthogonal to $\mathbf{v}$.

• Work and applications

  • Work done by a force $\mathbf{F}$ moving an object through displacement $\mathbf{d}$ with angle $\phi$ between them: $W = \mathbf{F}\cdot\mathbf{d}\;=\; |\mathbf{F}||\mathbf{d}|\cos\phi.$

• Vector operations and products

  • Cross product yields a vector normal to the plane; properties include anti-commutativity: $\mathbf{u}\times\mathbf{v} = -\mathbf{v}\times\mathbf{u}$.

  • Triple products and mixed products (scalar triple product, volume of a parallelepiped): $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$.

• Vector examples and practice

  • Examples illustrating addition, dot product, projection, and angle between vectors using $\cos\theta = (\mathbf{u}\cdot\mathbf{v})/(|\mathbf{u}||\mathbf{v}|)$ and related formulas.

Appendices, References, and Quick Reference (highlights)

• Appendix A: Summary of trigonometric formulas (quick reference for main identities; sine/cosine/tangent, Pythagorean identities, cofunctions, and a set of power-reducing, product-to-sum, and sum-to-product formulas).

• Appendix B: Solving the Ambiguous Case – Sine Validity Method (alternative process for SSA) including sine-value checks and angle determinations.

• Appendix C: Summary of Rectangular and Polar Forms (conversion tables between rectangular and polar coordinates).

• Appendix D: Trigonometry Reference Sheet (compact table of trig functions, conversion formulas, Euler’s formula, etc.).

Quick formulas to memorize (selected subset)

• Arc length: $s = r\theta,\quad \text{with } \theta\text{ in radians}.$

• Unit circle relations: $x=\cos\theta,\quad y=\sin\theta.$

• Reciprocal relations: $\csc\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta}.$

• Pythagorean identities: $\sin^2\theta+\cos^2\theta=1,\quad 1+\tan^2\theta=\sec^2\theta,\quad 1+\cot^2\theta=\csc^2\theta.$

• Sine/cosine/tangent for multiples: $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\quad \cos(A\pm B)=\cos A\cos B\mp\sin A\sin B,\quad \tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}.$

• Double-angle: $\sin(2A)=2\sin A\cos A,\quad \cos(2A)=\cos^2A-\sin^2A,\quad \tan(2A)=\frac{2\tan A}{1-\tan^2A}.$

• Half-angle (one form): $\sin\left(\frac{A}{2}\right)=\pm\sqrt{\frac{1-\cos A}{2}},\quad \cos\left(\frac{A}{2}\right)=\pm\sqrt{\frac{1+\cos A}{2}},\quad \tan\left(\frac{A}{2}\right)=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}.$

• Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$

• Law of Cosines: $a^2=b^2+c^2-2bc\cos A,\quad b^2=a^2+c^2-2ac\cos B,\quad c^2=a^2+b^2-2ab\cos C.$

• Area (triangle) formulas: $\text{Area} = \tfrac{1}{2}bh,\quad \text{Area} = \tfrac{1}{2}ab\sin C,\quad \text{Area}_{\text{Heron}} = \sqrt{s(s-a)(s-b)(s-c)},\; s=\tfrac{a+b+c}{2}.$

• Polar conversions: $x = r\cos\theta,\quad y = r\sin\theta,\quad r^2=x^2+y^2,\quad \theta=\tan^{-1}\left(\frac{y}{x}\right).$

• Euler/de Moivre basics: $z=re^{i\theta}=r(\cos\theta+i\sin\theta),\quad z^n=r^n(\cos n\theta + i\sin n\theta).$

• Vector dot/product essentials: $\mathbf{u}\cdot\mathbf{v}=u<em>xv</em>x+u<em>yv</em>y+u<em>zz</em>v,\quad \mathbf{u}\times \mathbf{v} = \det(\mathbf{i},\mathbf{j},\mathbf{k};u<em>x,u</em>y,u<em>z;v</em>x,v<em>y,v</em>z).$

• Projection: \text{proj}_{\mathbf{v}}\mathbf{u} = \left(\frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{v}|^2}\right)\mathbf{v}.$n- Work: W=\mathbf{F}\cdot\mathbf{d}=|\mathbf{F}||\mathbf{d}|\cos\phi.$$