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Every rule and concept in Trigonometry, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (165)

$\arccos(\cos\theta)$ is a “fold,” not the identity: Because $\arccos$ must output a value in $[0,\pi]$ , the composition $\arccos(\cos\theta)$ returns the unique number in $[0,\pi]$ having the same cosine as $\theta$ : \[ (θ)=; Inverse Trigonometric Functions › Going Deeper: Advanced Inverse Trig
$\arcsin x+\arccos x=\dfrac{\pi}{2}$: For every $x\in[-1,1]$ , \[ x+ x= $\frac{\pi}{2}$ .; Inverse Trigonometric Functions › Going Deeper: Advanced Inverse Trig
3D right-triangle problems: A three-dimensional figure is solved by finding a sequence of right triangles that lie in different planes but share an edge.; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
A circle of radius 1 that stores every angle: Draw a circle of radius $1$ centered at the origin and spin a point around it.; Core Ideas in Plain Terms › The Unit Circle, Plainly
Adding two arctangents: The tangent addition formula gives \[ ( a+ b)= $\frac{a+b}{1-ab}$ , \] so it is tempting to write $\arctan a+\arctan b=\arctan\!\dfrac{a+b}{1-ab}$ .; Inverse Trigonometric Functions › Going Deeper: Advanced Inverse Trig
Adding, subtracting, scaling, unit vectors: Vectors and the Dot Product › Vector Operations
Angles of elevation and depression: An angle of elevation is measured upward from a horizontal line to a line of sight; an angle of depression is measured downward from a horizontal line to a line of sight.; Right Triangle Trigonometry › Applications: Elevation, Depression, and Bearings
Apollonius Circles: the Locus $|z-a| = k\,|z-b|$: The set of points $z$ whose distance to $a$ is a fixed multiple $k>0$ of its distance to $b$ is:; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Arc Length and Sector Area: For a circle of radius $r$ and a central angle $\theta$ measured in radians: $ $\text{Arc length: } s = r\theta\qquad\qquad \text{Sector area: } A = \tfrac{1}{2}r^2\theta$ $If$ θ$ is given in degrees, convert to radians first.; Angles and Their Measure › Arc Length and Sector Area
Arc length, sector area, and speed: For a central angle $\theta$ measured in radians in a circle of radius $r$ : \[ s=rθ (arc length), A=12 r^2θ (sector area).; Course Review › Angles and Their Measure
Area Enclosed by a Polar Curve: The region swept by the ray to a polar curve $r = f(\theta)$ as $\theta$ runs from $\alpha$ to $\beta$ has area $ $\boxed{\ A = \frac12\int_{\alpha}^{\beta} r^2\,d\theta = \frac12\int_{\alpha}^{\beta} \bigl[f(\theta)\bigr]^2\,d\theta\ }$ $ This comes from sum…; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Area, revisited: five methods and area ratios: Depending on what you are given, any of these computes the area $K$ :; Law of Sines and Law of Cosines › Going Deeper: Advanced Triangle Laws
Bearings: A bearing such as $\text{N}60^\circ\text{E}$ names a direction by an acute angle measured from the north-south line toward the east or west.; Right Triangle Trigonometry › Applications: Elevation, Depression, and Bearings
Bearings and navigation with two legs: Give each leg of a trip as a bearing (measured from north) and a length.; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
Cevians: Stewart's Theorem, medians, and angle bisectors: A cevian is any segment from a vertex to the opposite side.; Law of Sines and Law of Cosines › Going Deeper: Advanced Triangle Laws
Chaining domain requirements: For a composition to be defined, the inner output must satisfy the outer function's domain and the inner function's own domain.; Inverse Trigonometric Functions › Going Deeper: Advanced Inverse Trig
Clock-Hand Angles: A clock face is $360^\circ$ , split into $12$ hours of $30^\circ$ each.; Angles and Their Measure › Going Deeper: Advanced Angle Ideas
Cofunction identities: In a right triangle the two acute angles are complementary (they sum to $90^\circ$ ).; Right Triangle Trigonometry › Cofunctions and the Complementary-Angle Relationship
Cofunction Identities (angles complementary): \[ (90^°-θ)=θ (90^°-θ)=θ (90^°-θ)=şcθ \] \[ (90^°-θ)=θ (90^°-θ)=θ\…; Trigonometric Identities › The Fundamental Identities
Combining exact values: products, sums, and powers: Once the special-angle coordinates are memorized, any expression built from them is just arithmetic with the numbers $0,\ \tfrac12,\ \tfrac{\sqrt2}{2},\ \tfrac{\sqrt3}{2},\ 1$ and their reciprocals.; The Unit Circle and Circular Functions › Going Deeper: Advanced Unit-Circle Ideas
Combining two sinusoids: $a\sin x+b\cos x=R\sin(x+\phi)$: A sum of a sine and a cosine of the same frequency is itself a single sinusoid.; Graphs of Sine and Cosine › Going Deeper: Advanced Sine & Cosine Graphs
Combining vectors in the real world: Resultant force / velocity: resolve each vector into components, add the components, then re-compute magnitude and direction.; Vectors and the Dot Product › Applications: Resultant Forces, Velocity, Navigation, Inclined Planes
Common decompositions: \[ 15^° = 45^°-30^°, 75^° = 45^°+30^°, 105^° = 60^°+45^°, $\tfrac{7\pi}{12}$ =105^°= $\tfrac{\pi}{3}$ + $\tfrac{\pi}{4}$ \] Then apply the matching sum or difference formula and simplify the radicals.; Sum, Difference, and Multiple-Angle Formulas › Finding Exact Values of Non-Special Angles
Common polar curves: Course Review › Polar Coordinates and Complex Numbers
Common Polar Curves: Polar Coordinates and Complex Numbers › Graphs of Polar Equations
Complements and Supplements: Two positive angles are; Angles and Their Measure › Complementary and Supplementary Angles
Components, magnitude, and direction: A vector $\mathbf{v}=\langle a,b\rangle$ from initial point $(x_1,y_1)$ to terminal point $(x_2,y_2)$ has components $\langle x_2-x_1,\ y_2-y_1\rangle$ .; Course Review › Vectors and the Dot Product
Composition rules: \[ (^-1x)=x for x∈[-1,1], ^-1(θ)=θ only if θ∈[- $\tfrac{\pi}{2}$ , $\tfrac{\pi}{2}$ ].; Course Review › Inverse Trigonometric Functions
Conditional identities from $\tan x$: If you are given $\tan x=t$ , divide numerator and denominator by $\cos^2 x$ to turn symmetric expressions into rational functions of $t$ : \[ x x= $\frac{\tan x}{1+\tan^2 x}$ = $\frac{t}{1+t^2}$ , \] \[ ^2 x-^2 x= $\frac{\tan^2 x-1}{\tan^2 x+1}$ =…; Trigonometric Identities › Going Deeper: Advanced Identities
Cotangent: key features: Since $\cot x=\dfrac{\cos x}{\sin x}$ , the graph has a vertical asymptote wherever $\sin x=0$ .; Graphs of the Other Trigonometric Functions › The Graph of $y=\cot x$
Coterminal Angles: Two angles are coterminal if they share the same terminal side.; Angles and Their Measure › Coterminal Angles
Counting solutions of a multiple-angle equation: For $\sin(kx)=c$ or $\cos(kx)=c$ with $k$ a positive integer and $-1<c<1$ , the whole angle $kx$ ranges over $[0,2\pi k)$ when $x\in[0,2\pi)$ .; Trigonometric Equations › Going Deeper: Advanced Trig Equations
Definition from a point on the circle: The unit circle is the circle $x^2+y^2=1$ centered at the origin with radius $1$ .; The Unit Circle and Circular Functions › The Unit Circle and the Six Functions
Definition from the unit circle: On the unit circle $x^2+y^2=1$ , if the terminal side of $\theta$ meets the circle at $(x,y)$ then \[ θ=x, θ=y, θ= $\frac{y}{x}$ , \] with $\sec\theta=\frac1x$ , $\csc\theta=\frac1y$ , $\cot\theta=\frac xy$ .; Course Review › The Unit Circle and Circular Functions
Definitions from a right triangle: For an acute angle $\theta$ in a right triangle, label the sides relative to $\theta$ : the opposite leg (across from $\theta$ ), the adjacent leg (next to $\theta$ , not the hypotenuse), and the hypotenuse (across from the right angle).; Right Triangle Trigonometry › The Six Trigonometric Ratios (SOH-CAH-TOA)
Degrees, Minutes, Seconds: One full revolution is $360^\circ$ .; Angles and Their Measure › Degree Measure and DMS
Degrees, radians, and conversion: An angle in standard position has its vertex at the origin and its initial side on the positive $x$ -axis.; Course Review › Angles and Their Measure
DeMoivre's Theorem (Powers): For $z = r(\cos\theta + i\sin\theta)$ and any positive integer $n$ : $ $\boxed{\,z^n = r^n\big(\cos n\theta + i\sin n\theta\big)\,}$ $Raise the modulus to the power; multiply the angle by$ n$.; Polar Coordinates and Complex Numbers › DeMoivre's Theorem: Powers and Roots
Direction of an angle bisector: The bisector of the angle between $\vec u$ and $\vec v$ points along the sum of the two unit vectors: \[ b= u+ v= $\frac{\vec u}{\|\vec u\|}$ + $\frac{\vec v}{\|\vec v\|}$ .; Vectors and the Dot Product › Going Deeper: Advanced Vector Ideas
Domains and ranges of the inverses: Course Review › Inverse Trigonometric Functions
Dot product and the angle between vectors: \[ u· v= a,b· c,d=ac+bd (a scalar, not a vector).; Vectors and the Dot Product › The Dot Product, Angle Between Vectors, Orthogonality
Dot product, angle, projection, work: For $\mathbf{u}=\langle u_1,u_2\rangle$ and $\mathbf{v}=\langle v_1,v_2\rangle$ : \[ u·v=u_1v_1+u_2v_2=\|u\| \|v\|θ.; Course Review › Vectors and the Dot Product
Double-angle and half-angle: Course Review › Sum, Difference, and Multiple-Angle Formulas
Double-angle identities: \[ 2A = 2 A A, 2A = $\dfrac{2\tan A}{1-\tan^2 A}$ \] \[ 2A = ^2 A - ^2 A = 2^2 A - 1 = 1 - 2^2 A \] Choose the cosine form that matches what you already know: use $2\cos^2A-1$ when you know $\cos A$ , and $1-2^2…; Sum, Difference, and Multiple-Angle Formulas › Double-Angle Formulas
Draw, label, choose the tool: Sketch the triangle and label everything you know.; Core Ideas in Plain Terms › Problem-Solving Playbook
Equally spaced angles: the cosines (and sines) sum to zero: If you place $n\ge 2$ points evenly around the unit circle, their $x$ -coordinates cancel and so do their $y$ -coordinates: \[ ∑_k=0^n-1 (θ+ $\dfrac{2\pi k}{n}$ )=0, ∑_k=0^n-1 (θ+ $\dfrac{2\pi k}{n}$ )=0.; The Unit Circle and Circular Functions › Going Deeper: Advanced Unit-Circle Ideas
Equation Conversion Toolkit: Swap between forms with the same relationships, plus these handy substitutions: $ $r^2 = x^2+y^2,\qquad x = r\cos\theta,\qquad y = r\sin\theta,\qquad \tan\theta = \frac{y}{x}$ $Polar$ → $rectangular: multiply by$ r $or use identities to remove$ r $and$ …; Polar Coordinates and Complex Numbers › Converting Polar and Rectangular Equations
Equilibrium: resultant of several forces is zero: An object is in equilibrium when the forces on it sum to the zero vector: \[ ∑ F= 0 ∑ F_x=0 and ∑ F_y=0 .; Vectors and the Dot Product › Going Deeper: Advanced Vector Ideas
Even and odd functions: Reflecting an angle to $-\theta$ reflects the point $(x,y)$ across the $x$ -axis to $(x,-y)$ .; The Unit Circle and Circular Functions › Even and Odd (Symmetry) Properties
Even/Odd (Negative-Angle) Identities: Even (unchanged by a sign flip): \[ (-θ)=θ (-θ)=θ \] Odd (pick up a negative sign): \[ (-θ)=-θ (-θ)=-θ şc(-θ)=-şcθ (-θ)=-θ \]; Trigonometric Identities › The Fundamental Identities
Factoring, quadratic form, and multiple angles: Course Review › Trigonometric Equations
Five steps for any sinusoid: Graphs of Sine and Cosine › Graphing the Full Sinusoid $y=a\sin(bx-c)+d$
Full analysis of $y=a\sec(bx-c)+d$: Sketch the guide wave $y=a\cos(bx-c)+d$ first; the secant hugs its humps.; Graphs of the Other Trigonometric Functions › Going Deeper: Advanced Graph Ideas
Full analysis of $y=a\tan(bx-c)+d$: Treat the whole function as one transformed tangent.; Graphs of the Other Trigonometric Functions › Going Deeper: Advanced Graph Ideas
Gear, Pulley, and Belt Chains: When two wheels are connected, one of two rules applies:; Angles and Their Measure › Going Deeper: Advanced Angle Ideas
General solutions: Solve for the reference angle, place it in every quadrant the sign allows, then add the period: \[ ,,,şc: add +2π n, ,: add +π n, \] where $n$ is any integer.; Course Review › Trigonometric Equations
Half-angle identities: \[ $\frac{A}{2}$ = ± $\dfrac{1-\cos A}{2}$ , $\frac{A}{2}$ = ± $\dfrac{1+\cos A}{2}$ \] \[ $\frac{A}{2}$ = $\frac{1-\cos A}{\sin A}$ = $\frac{\sin A}{1+\cos A}$ = ± $\dfrac{1-\cos A}{1+\cos A}$ \] The $\pm$ is decided by the quadran…; Sum, Difference, and Multiple-Angle Formulas › Half-Angle Formulas
Intersections and counting solutions on an interval: To find where two graphs meet, or how many times an equation is satisfied on an interval, think geometrically:; Graphs of Sine and Cosine › Going Deeper: Advanced Sine & Cosine Graphs
Intersections of Polar Curves: To find where $r = f(\theta)$ and $r = g(\theta)$ meet, solving $f(\theta) = g(\theta)$ is not enough.; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Law of Cosines: For any triangle $ABC$ , \[ a^2=b^2+c^2-2bc A, b^2=a^2+c^2-2ac B, c^2=a^2+b^2-2ab C.; Law of Sines and Law of Cosines › The Law of Cosines (SAS and SSS)
Law of Sines: For any triangle $ABC$ , \[ $\frac{a}{\sin A}$ = $\frac{b}{\sin B}$ = $\frac{c}{\sin C}$ .; Law of Sines and Law of Cosines › The Law of Sines (AAS and ASA)
Lowering even powers: \[ ^2 A = $\frac{1-\cos 2A}{2}$ , ^2 A = $\frac{1+\cos 2A}{2}$ , ^2 A = $\frac{1-\cos 2A}{1+\cos 2A}$ \] These are just the double-angle cosine identity solved for $\sin^2A$ and $\cos^2A$ .; Sum, Difference, and Multiple-Angle Formulas › Power-Reducing Formulas
Magnitude and direction angle: \[ \| v\|=\| a,b\|= $\sqrt{a^2+b^2}$ , θ= $\frac{b}{a}$ .; Vectors and the Dot Product › Vectors: Geometric and Component Form
Magnitude of a sum or difference from the angle: You do not need components to find $\|\vec u+\vec v\|$ — only the two magnitudes and the angle $\theta$ between the vectors.; Vectors and the Dot Product › Going Deeper: Advanced Vector Ideas
Many Names for One Point: Every polar point has infinitely many representations: $ $(r,\theta) = (r,\ \theta + 360^\circ k) = (-r,\ \theta + 180^\circ + 360^\circ k),\qquad k = 0,\pm1,\pm2,\dots$ $The pole itself is$ (0,θ) $for any angle$ θ$.; Polar Coordinates and Complex Numbers › Polar Coordinates
Max, min, and the period of a sum: Once a sinusoid is written as $y=A\sin(bx-c)+d$ (or found via $R$ above), reading its extremes is mechanical:; Graphs of Sine and Cosine › Going Deeper: Advanced Sine & Cosine Graphs
Multiply and Divide by Modulus and Angle: For $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$ : $ $z_1 z_2 = r_1 r_2\big[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big]$ $ $\frac{z_1}{z_2}$ = $\frac{r_1}{r_2}$ [(θ_1-θ_2) + i(θ_1-\…; Polar Coordinates and Complex Numbers › Products and Quotients in Trig Form
Nested half-angles for unusual exact values: Applying the half-angle cosine formula repeatedly to $\cos\tfrac{\pi}{4}=\tfrac{\sqrt2}{2}$ builds a tower of nested radicals: \[ $\frac{\pi}{8}$ = 1+ $\tfrac{\pi}{4}$ 2 = 12 $\sqrt{2+\sqrt2}$ , $\frac{\pi}{16}$ = 122+…; Sum, Difference, and Multiple-Angle Formulas › Going Deeper: Advanced Formula Techniques
One value plus a sign condition (the twist): A classic problem gives one function value together with a sign clue instead of a named quadrant, and often the value is a reciprocal function.; The Unit Circle and Circular Functions › Going Deeper: Advanced Unit-Circle Ideas
Optimizing a viewing angle: A picture (or screen) hangs on a wall with its bottom edge $a$ above eye level and its top edge $b$ above eye level.; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
Parameter conditions: how many solutions does $\sin x = k$ have?: Think of solving $f(x)=k$ as intersecting the graph $y=f(x)$ with the horizontal line $y=k$ .; Trigonometric Equations › Going Deeper: Advanced Trig Equations
Parameterized terminal points: Writing $P(\theta)=(\cos\theta,\ \sin\theta)$ turns the unit circle into a machine: feed in the angle, read out the terminal point.; The Unit Circle and Circular Functions › Going Deeper: Advanced Unit-Circle Ideas
Period $=\dfrac{2\pi}{b}$: In $y=a\sin(bx)$ or $y=a\cos(bx)$ (with $b>0$ ), the coefficient $b$ controls how fast the wave cycles: $ $\boxed{\ \text{Period}=\dfrac{2\pi}{b}\ }\qquad\text{so}\qquad b=\dfrac{2\pi}{\text{Period}}.$ $; Graphs of Sine and Cosine › Period and the Coefficient $b$
Period, phase shift, and asymptotes: For $b>0$ , write the inside as $b\left(x-\dfrac{c}{b}\right)$ .; Graphs of the Other Trigonometric Functions › Transformations: $y=a\,f(bx-c)+d$
Polar $\leftrightarrow$ Rectangular (Points): With the pole at the origin and polar axis along the positive $x$ -axis: $ $\boxed{\,x = r\cos\theta,\qquad y = r\sin\theta\,}\qquad\qquad \boxed{\,r = \sqrt{x^2+y^2},\qquad \tan\theta = \dfrac{y}{x}\,}$ $When finding$ θ $, always check the quadrant of$ (…; Polar Coordinates and Complex Numbers › Polar Coordinates
Polar and rectangular conversion: A point $(r,\theta)$ in polar coordinates relates to rectangular $(x,y)$ by \[ x=rθ, y=rθ, \] \[ r^2=x^2+y^2, θ= $\frac{y}{x}$ (choose θ by quadrant).; Course Review › Polar Coordinates and Complex Numbers
Power-reducing, product-to-sum, sum-to-product: Power-reducing: \[ ^2 A= $\frac{1-\cos 2A}{2}$ , ^2 A= $\frac{1+\cos 2A}{2}$ , ^2 A= $\frac{1-\cos 2A}{1+\cos 2A}$ .; Course Review › Sum, Difference, and Multiple-Angle Formulas
Power-Sum Reductions (memorize the pattern): Since $s^2+c^2=1$ , every even power sum collapses to a polynomial in $p=sc$ : \[ ^4 x+^4 x=1-2p^2, \] \[ ^6 x+^6 x=1-3p^2, \] \[ ^4 x+^4 x = 1-2^2 x^2 x, ^6 x+^6 x = 1-3^2 x^2 x.; Trigonometric Identities › Going Deeper: Advanced Identities
Projection and parallel/orthogonal decomposition: The (vector) projection of $\vec u$ onto $\vec v$ : \[ proj_ v u=( $\frac{\vec u\cdot\vec v}{\|\vec v\|^2}$ ) v .; Vectors and the Dot Product › Vector Projection and Work
Proving a relationship: $a^2=b(b+c)\ \Rightarrow\ A=2B$: Advanced problems often ask you to prove a triangle identity rather than solve for a number.; Law of Sines and Law of Cosines › Going Deeper: Advanced Triangle Laws
Proving geometry with vectors: Turn a geometric claim about points $A,B,C,\dots$ into an algebra statement about the vectors between them:; Vectors and the Dot Product › Going Deeper: Advanced Vector Ideas
Pythagorean Identities: \[ ^2θ+^2θ=1 \] \[ 1+^2θ=^2θ 1+^2θ=şc^2θ \] Each has useful rearrangements, e.g. $\sin^2\theta=1-\cos^2\theta$ and $\tan^2\theta=\sec^2\theta-1$ .; Trigonometric Identities › The Fundamental Identities
Quadrantal Angles: A quadrantal angle has its terminal side lying on an axis.; Angles and Their Measure › Angles in Standard Position
Quotient Identities: \[ θ= $\frac{\sin\theta}{\cos\theta}$ θ= $\frac{\cos\theta}{\sin\theta}$ \]; Trigonometric Identities › The Fundamental Identities
Radians: One radian is the central angle that subtends an arc equal in length to the radius.; Angles and Their Measure › Radian Measure and Conversions
Radians measure angles by arc length: Instead of $360^\circ$ for a full turn, radians use $2\pi$ — the distance around a radius- $1$ circle.; Core Ideas in Plain Terms › Radians and Wave Graphs
Reading a proof: Do not read a proof the way you read a rule.; Law of Sines and Law of Cosines › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Vectors and the Dot Product › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Polar Coordinates and Complex Numbers › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Angles and Their Measure › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Right Triangle Trigonometry › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; The Unit Circle and Circular Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Graphs of Sine and Cosine › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Graphs of the Other Trigonometric Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Inverse Trigonometric Functions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Trigonometric Identities › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Sum, Difference, and Multiple-Angle Formulas › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Trigonometric Equations › Why It Works
Reading a sine graph: For $y = a\sin(bx)$ , the amplitude $|a|$ is how tall the wave gets, and the period $\tfrac{2\pi}{b}$ is how wide one full wave is before it repeats.; Core Ideas in Plain Terms › Radians and Wave Graphs
Reciprocal Identities: \[ şcθ= $\frac{1}{\sin\theta}$ θ= $\frac{1}{\cos\theta}$ θ= $\frac{1}{\tan\theta}$ \] \[ θ= $\frac{1}{\csc\theta}$ θ= $\frac{1}{\sec\theta}$ θ= $\frac{1}{\cot\theta}$ \]; Trigonometric Identities › The Fundamental Identities
Reference angle: The reference angle $\theta'$ is the acute angle between the terminal side and the $x$ -axis.; The Unit Circle and Circular Functions › Reference Angles and Signs by Quadrant
Reference angles, signs, period, and symmetry: The reference angle $\theta'$ is the acute angle to the $x$ -axis: \[ Q II: 180^°-θ, Q III: θ-180^°, Q IV: 360^°-θ.; Course Review › The Unit Circle and Circular Functions
Resolving into components: A vector with magnitude $r=\|\vec v\|$ and direction angle $\theta$ has components \[ v= rθ, rθ .; Vectors and the Dot Product › Writing a Vector from Magnitude and Direction
Reverse-engineering a sinusoid: Given a graph or a word description, recover $a,b,c,d$ :; Graphs of Sine and Cosine › Writing an Equation from a Graph or Description
Revolutions per Minute and Unit Conversions: Rotating machinery is usually rated in RPM (revolutions per minute).; Angles and Their Measure › Going Deeper: Advanced Angle Ideas
Right triangles inside inscribed figures: Two facts turn geometry problems into right-triangle trig:; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
Roots of Unity: The $n$ th roots of unity are the $n$ solutions of $z^n = 1$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
Secant and cosecant: \[ y=a(b(x-c)) and y=aşc(b(x-c)): period= $\frac{2\pi}{b}$ .; Course Review › Graphs of the Other Trig Functions
Secant and cosecant as reciprocals: Because $\sec x=\dfrac{1}{\cos x}$ and $\csc x=\dfrac{1}{\sin x}$ , you can sketch each one from its “parent” wave:; Graphs of the Other Trigonometric Functions › The Graphs of $y=\sec x$ and $y=\csc x$
Sector Systems with Two Solutions: A sector is fixed by any two of $\{r,\ \theta,\ s,\ A\}$ .; Angles and Their Measure › Going Deeper: Advanced Angle Ideas
Setting up word problems: Law of Sines and Law of Cosines › Applications: Surveying, Navigation, and Bearings
Shifting the graph: $y=a\sin(bx-c)+d$: Two more numbers slide the whole graph:; Graphs of Sine and Cosine › Phase (Horizontal) Shift and Vertical Shift
Signs by quadrant (ASTC): “All Students Take Calculus”: the function that is positive in QI, QII, QIII, QIV.; Trigonometric Identities › Finding the Other Five Values from One
Sine, cosine, tangent are just fractions: In a right triangle, pick one of the non-right angles.; Core Ideas in Plain Terms › What the Trig Ratios Really Are
SOH-CAH-TOA and reciprocals: For an acute angle $\theta$ in a right triangle, \[ θ=opphyp, θ=adjhyp, θ=oppadj.; Course Review › Right Triangle Trigonometry
Special angles and cofunctions: Course Review › Right Triangle Trigonometry
SSA: how many triangles?: Given angle $A$ , its opposite side $a$ , and an adjacent side $b$ .; Law of Sines and Law of Cosines › The Ambiguous Case (SSA)
Standard Position: An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive $x$ -axis.; Angles and Their Measure › Angles in Standard Position
Strategies: Trigonometric Identities › Verifying Trigonometric Identities
Strategy: Sum, Difference, and Multiple-Angle Formulas › Simplifying, Verifying Identities, and Finding Values
Sum and difference: \[ (A± B)= A B± A B \] \[ (A± B)= A B∓ A B \] \[ (A± B)= $\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$ \] Cosine flips the sign; sine and tangent keep it.; Course Review › Sum, Difference, and Multiple-Angle Formulas
Sum-to-product and double-angle as an opening move: Some equations cannot be factored until you first merge or split terms.; Trigonometric Equations › Going Deeper: Advanced Trig Equations
Summing equally spaced sines and cosines: For an arithmetic progression of angles $a,\,a+d,\,a+2d,\dots$ , multiply by $2\sin\tfrac{d}{2}$ and telescope with product-to-sum.; Sum, Difference, and Multiple-Angle Formulas › Going Deeper: Advanced Formula Techniques
Symmetric-Function Toolkit: Write $s=\sin x$ , $c=\cos x$ .; Trigonometric Identities › Going Deeper: Advanced Identities
Systems of trig equations: To solve two trig equations simultaneously, find each variable's candidate set, then keep only the pairs that satisfy both.; Trigonometric Equations › Going Deeper: Advanced Trig Equations
Table of exact values: \[; Right Triangle Trigonometry › Exact Values of the Special Angles $30^\circ,45^\circ,60^\circ$
Tangent and cotangent: \[ y=a(b(x-c)): period= $\frac{\pi}{b}$ , asymptotes where bx= $\tfrac{\pi}{2}$ +π n.; Course Review › Graphs of the Other Trig Functions
Tangent: key features: Since $\tan x=\dfrac{\sin x}{\cos x}$ , the graph has a vertical asymptote wherever $\cos x=0$ .; Graphs of the Other Trigonometric Functions › The Graph of $y=\tan x$
Telescoping products of cosines: Multiplying a chain of angle-doubling cosines by $2\sin\theta$ collapses it, because each step obeys $2\sin\phi\cos\phi=\sin 2\phi$ .; Sum, Difference, and Multiple-Angle Formulas › Going Deeper: Advanced Formula Techniques
The $n$ Distinct $n$ th Roots: Every nonzero $z = r(\cos\theta + i\sin\theta)$ has exactly $n$ complex $n$ th roots: $ $w_k = \sqrt[n]{r}\left[\cos\!\left(\frac{\theta + 360^\circ k}{n}\right) + i\sin\!\left(\frac{\theta + 360^\circ k}{n}\right)\right],\qquad k = 0,1,2,\dots,n-1.$ $ All roo…; Polar Coordinates and Complex Numbers › DeMoivre's Theorem: Powers and Roots
The $R$ -form: solving $a\sin x + b\cos x = c$: A sum of a sine and a cosine with the same angle can be rewritten as a single sinusoid: \[ a x + b x = R(x+φ), R= $\sqrt{a^2+b^2}$ , φ = $\frac{b}{a}$ .; Trigonometric Equations › Going Deeper: Advanced Trig Equations
The altitude-on-hypotenuse geometric mean: Drop the altitude of length $h$ from the right angle to the hypotenuse.; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
The Complex Plane: A complex number $z = a + bi$ is plotted as the point $(a,b)$ : the real part $a$ on the horizontal axis, the imaginary part $b$ on the vertical axis.; Polar Coordinates and Complex Numbers › Complex Numbers and Trigonometric Form
The Extended Law of Sines: circumradius and inradius: The common ratio in the Law of Sines is not just a number—it equals the diameter of the circumscribed circle (the circle through all three vertices, radius $R$ ): \[ $\frac{a}{\sin A}$ = $\frac{b}{\sin B}$ = $\frac{c}{\sin C}$ =2R.; Law of Sines and Law of Cosines › Going Deeper: Advanced Triangle Laws
The fundamental identities: Course Review › Trigonometric Identities
The general sinusoid: For $y=a\sin\big(b(x-c)\big)+d$ or $y=a\cos\big(b(x-c)\big)+d$ with $b>0$ : \[ amplitude=|a|, period= $\frac{2\pi}{b}$ , \] \[ phase shift=c (right if c>0), vertical shift=d.; Course Review › Graphs of Sine and Cosine
The idea: To solve $\sin x = k$ (or $\cos x = k$ , $\tan x = k$ ):; Trigonometric Equations › Solving Basic Trigonometric Equations
The Identity $z^n + z^{-n} = 2\cos n\theta$: If $z = \cos\theta + i\sin\theta$ lies on the unit circle, then $\tfrac1z = \bar z = \cos\theta - i\sin\theta$ .; Polar Coordinates and Complex Numbers › Going Deeper: Advanced Polar & Complex Ideas
The Law of Sines, plainly: In any triangle, $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ : the bigger an angle, the bigger the side across from it, in exact proportion.; Core Ideas in Plain Terms › Going Deeper: Identities and the Law of Sines
The parent sine and cosine curves: Both $y=\sin x$ and $y=\cos x$ are periodic with period $2\pi$ : the graph repeats every $2\pi$ units.; Graphs of Sine and Cosine › The Basic Graphs $y=\sin x$ and $y=\cos x$
The Polar Point $(r,\theta)$: A polar point is written $(r,\theta)$ , where $r$ is the directed distance from the origin (the pole) and $\theta$ is the angle from the positive $x$ -axis (the polar axis).; Polar Coordinates and Complex Numbers › Polar Coordinates
The role of $a$ in $y=a\sin x$ and $y=a\cos x$: The number $a$ stretches the graph vertically.; Graphs of Sine and Cosine › Amplitude $|a|$ and Vertical Reflection
The six circular functions at a glance: \[; The Unit Circle and Circular Functions › Domain, Range, and Periodicity
The six core formulas: \[; Sum, Difference, and Multiple-Angle Formulas › Sum and Difference Formulas
The special-angle circle: The special angles are multiples of $30^\circ=\tfrac{\pi}{6}$ and $45^\circ=\tfrac{\pi}{4}$ .; The Unit Circle and Circular Functions › Exact Values at the Special Angles (All Four Quadrants)
The three inverse trig functions: $y=\arcsin x$ means “ $y$ is the angle whose sine is $x$ ”: $\sin y = x$ .; Inverse Trigonometric Functions › Definitions: Domain & Range
The two laws: For any triangle with sides $a,b,c$ opposite angles $A,B,C$ : \[ Law of Sines: $\frac{a}{\sin A}$ = $\frac{b}{\sin B}$ = $\frac{c}{\sin C}$ .; Course Review › Law of Sines and Law of Cosines
The two special triangles: Every special-angle value comes from two triangles.; Right Triangle Trigonometry › Exact Values of the Special Angles $30^\circ,45^\circ,60^\circ$
The universal rewrite: \[ θ= $\frac{\sin\theta}{\cos\theta}$ θ= $\frac{\cos\theta}{\sin\theta}$ θ= $\frac{1}{\cos\theta}$ şcθ= $\frac{1}{\sin\theta}$ \]; Trigonometric Identities › Rewriting in Terms of Sine and Cosine (or as One Function)
Transformations that produce identical graphs: Different-looking formulas can trace the exact same curve.; Graphs of the Other Trigonometric Functions › Going Deeper: Advanced Graph Ideas
Trig form, DeMoivre, and $n$ th roots: A complex number $z=a+bi$ has trig (polar) form \[ z=r(θ+iθ), r= $\sqrt{a^2+b^2}$ , θ= ba.; Course Review › Polar Coordinates and Complex Numbers
Trigonometric (Polar) Form: $ $z = a + bi = r(\cos\theta + i\sin\theta),\qquad \text{where } a = r\cos\theta,\ \ b = r\sin\theta.$ $The shorthand$ r(θ + iθ) $is often written$ r cis θ$.; Polar Coordinates and Complex Numbers › Complex Numbers and Trigonometric Form
Triple-angle formulas: Chaining a double-angle with a sum formula ( $3A=2A+A$ ) gives the triple-angle identities: \[ 3A = 3 A - 4^3 A, 3A = 4^3 A - 3 A, \] \[ 3A = $\frac{3\tan A - \tan^3 A}{1-3\tan^2 A}$ .; Sum, Difference, and Multiple-Angle Formulas › Going Deeper: Advanced Formula Techniques
Turning a sum into a difference: The difference version of the formula is often exact with $k=0$ : \[ $\frac{1}{n}$ - $\frac{1}{n+1}$ = $\frac{1}{1+n(n+1)}$ = $\frac{1}{n^2+n+1}$ .; Inverse Trigonometric Functions › Going Deeper: Advanced Inverse Trig
Two area formulas: SAS (two sides and the included angle): \[ Area=12 ab C=12 bc A=12 ac B.; Law of Sines and Law of Cosines › Area of a Triangle
Two families of identities: Product-to-sum: \[; Sum, Difference, and Multiple-Angle Formulas › Product-to-Sum and Sum-to-Product
Two Kinds of Speed: Suppose a point moves along a circle of radius $r$ , sweeping angle $\theta$ (radians) in time $t$ .; Angles and Their Measure › Linear Speed and Angular Speed
Two observers / two towers (shared height): When two observers on the same horizontal line sight the same point, both right triangles share the vertical height $h$ .; Right Triangle Trigonometry › Going Deeper: Advanced Right-Triangle Trig
Using a calculator: For values that are not special angles, use your calculator (in degree mode).; Right Triangle Trigonometry › Calculator Values and Solving Right Triangles
What a vector is: A vector has both magnitude (length) and direction.; Vectors and the Dot Product › Vectors: Geometric and Component Form
When the point is not on the unit circle: If the terminal side passes through any point $(x,y)\neq(0,0)$ , let \[ r= $\sqrt{x^2+y^2}$ >0.; The Unit Circle and Circular Functions › Evaluating from a Point on the Terminal Side
Which law, plus area: Course Review › Law of Sines and Law of Cosines
Why $\sin^2\theta + \cos^2\theta = 1$: The unit-circle point $(\cos\theta, \sin\theta)$ sits exactly $1$ away from the origin, and distance is $\sqrt{x^2 + y^2}$ .; Core Ideas in Plain Terms › Going Deeper: Identities and the Law of Sines
Work done by a force: When a constant force $\vec F$ moves an object through displacement $\vec d$ , \[ W= F· d=\| F\| \| d\|θ, \] where $\theta$ is the angle between the force and the direction of motion.; Vectors and the Dot Product › Vector Projection and Work
Writing a sinusoid from scattered conditions, and identical graphs: From scattered data.; Graphs of Sine and Cosine › Going Deeper: Advanced Sine & Cosine Graphs