Trigonometry - Unit 10 Formula Sheet

Core Principles of Trigonometric Identities

  • Fundamental Reduction: Most trigonometric identities reduce directly to three foundational mathematical facts:
    • sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
    • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
    • cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
  • Quadrant Sign Dependence: The algebraic sign of any trigonometric ratio must always be evaluated and maintained according to the specific quadrant in which the angle terminates.

Definitions and the Six Trigonometric Functions

  • Right Triangle Geometry:
    • Defined for an acute angle θ\theta situated in a right-angled triangle.
    • Triangle side designations:
    • PP = Perpendicular side (the side Opposite to θ\theta)
    • BB = Base side (the side Adjacent to θ\theta)
    • HH = Hypotenuse side (the longest side, opposite the right angle)
    • OO = Opposite side (PP)
    • AA = Adjacent side (BB)
  • SOH — CAH — TOA Primary Ratios:
    • SOH: sin⁡(θ)=OH  ⟹  sin⁡(θ)=OppositeHypotenuse=PH\sin(\theta) = \frac{O}{H} \implies \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{P}{H}
    • CAH: cos⁡(θ)=AH  ⟹  cos⁡(θ)=AdjacentHypotenuse=BH\cos(\theta) = \frac{A}{H} \implies \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{B}{H}
    • TOA: tan⁡(θ)=OA  ⟹  tan⁡(θ)=OppositeAdjacent=PB\tan(\theta) = \frac{O}{A} \implies \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{P}{B}
  • Reciprocal Relationships:
    • Pairings:
    • sin⁡↔csc⁡\sin \leftrightarrow \csc
    • cos⁡↔sec⁡\cos \leftrightarrow \sec
    • tan⁡↔cot⁡\tan \leftrightarrow \cot
    • Operational meaning: Reciprocal denotes inverting (flipping) the geometric ratio, such that OH→HO\frac{O}{H} \rightarrow \frac{H}{O} (\text{OH} \rightarrow \text{HO}$).\n * Functional pairings:\n * \sin(\theta) = \frac{P}{H}withreciprocalwith reciprocal\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{H}{P}\n * \cos(\theta) = \frac{B}{H}withreciprocalwith reciprocal\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{H}{B}\n * \tan(\theta) = \frac{P}{B}withreciprocalwith reciprocal\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{B}{P}\n * \cot(\theta) = \frac{B}{P}withreciprocalwith reciprocal\tan(\theta) = \frac{1}{\cot(\theta)}\n * \sec(\theta) = \frac{H}{B}withreciprocalwith reciprocal\cos(\theta) = \frac{1}{\sec(\theta)}\n * \csc(\theta) = \frac{H}{P}withreciprocalwith reciprocal\sin(\theta) = \frac{1}{\csc(\theta)}\n* **Functions Expressed in Terms of Sine and Cosine:**\n * \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\n * \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\n * \sec(\theta) = \frac{1}{\cos(\theta)}\n * \csc(\theta) = \frac{1}{\sin(\theta)}\n\n# Fundamental Pythagorean, Reciprocal, and Quotient Identities\n\n* **Pythagorean Identities:**\n * Standard forms:\n * \sin^2(\theta) + \cos^2(\theta) = 1\n * 1 + \tan^2(\theta) = \sec^2(\theta)\n * 1 + \cot^2(\theta) = \csc^2(\theta)\n * Useful algebraic rearrangements:\n * \sin^2(\theta) = 1 - \cos^2(\theta)\n * \cos^2(\theta) = 1 - \sin^2(\theta)\n * \tan^2(\theta) = \sec^2(\theta) - 1\n * \cot^2(\theta) = \csc^2(\theta) - 1\n* **Reciprocal and Quotient Product Identities:**\n * \sin(\theta)\csc(\theta) = 1\n * \cos(\theta)\sec(\theta) = 1\n * \tan(\theta)\cot(\theta) = 1\n * \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)\n * \frac{\cos(\theta)}{\sin(\theta)} = \cot(\theta)\n* **Equivalent Equation Forms for Simplification:**\n * \tan(\theta)\cot(\theta) - 1 = 0\n * \sec(\theta)\cos(\theta) - 1 = 0\n * \csc(\theta)\sin(\theta) - 1 = 0\n * \sin^2(\theta) + \cos^2(\theta) - 1 = 0\n * \sec^2(\theta) - \tan^2(\theta) - 1 = 0\n * \csc^2(\theta) - \cot^2(\theta) - 1 = 0\n\n# Fundamental Law and Compound-Angle Identities\n\n* **The Fundamental Law of Trigonometry:**\n * Base equation:\n * \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\n * Derivation role: All compound-angle sum and difference identities are derived directly from this single fundamental equation.\n* **Compound-Angle Identities (Addition and Subtraction):**\n * **Sine Compound Formulas:**\n * \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\n * \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\n * **Cosine Compound Formulas:**\n * \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\n * \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\n * **Tangent Compound Formulas:**\n * \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}\n * \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\n\n# Allied Angles and Transformation Rules\n\n* **Transformation Rules:**\n * **Odd Multiples of 90^\circ:∗∗Ifanangle:** If an angle\thetaisaddedtoorsubtractedfromanoddmultipleofis added to or subtracted from an odd multiple of90^\circ(suchas(such as90^\circ \pm \thetaoror270^\circ \pm \theta), the trigonometric ratio transitions into its complementary co-ratio:\n * \sin \leftrightarrow \cos\n * \tan \leftrightarrow \cot\n * \sec \leftrightarrow \csc\n * **Even Multiples of 90^\circ:∗∗Ifanangle:** If an angle\thetaisaddedtoorsubtractedfromanevenmultipleofis added to or subtracted from an even multiple of90^\circ(suchas(such as180^\circ \pm \thetaoror360^\circ \pm \theta), the ratio type remains unchanged.\n * **Sign Rule:** The positive or negative algebraic sign is determined exclusively by the quadrant in which the compound angle terminates.\n* **Sine Transformations Across All Forms:**\n * 90^\circ - \theta:co−ratio,: co-ratio,\sin(90^\circ - \theta) = \cos(\theta)\n * 90^\circ + \theta:co−ratio,: co-ratio,\sin(90^\circ + \theta) = \cos(\theta)\n * 180^\circ - \theta:sameratio,: same ratio,\sin(180^\circ - \theta) = \sin(\theta)\n * 180^\circ + \theta:sameratio,: same ratio,\sin(180^\circ + \theta) = -\sin(\theta)\n * 270^\circ - \theta:co−ratio,: co-ratio,\sin(270^\circ - \theta) = -\cos(\theta)\n * 270^\circ + \theta:co−ratio,: co-ratio,\sin(270^\circ + \theta) = -\cos(\theta)\n * 360^\circ - \theta:sameratio,: same ratio,\sin(360^\circ - \theta) = -\sin(\theta)\n * 360^\circ + \theta:sameratio,: same ratio,\sin(360^\circ + \theta) = \sin(\theta)\n* **Comprehensive Allied-Angle Table:**\n * **Negative Angle (-\theta):**\n * \sin(-\theta) = -\sin(\theta)\n * \cos(-\theta) = \cos(\theta)\n * \tan(-\theta) = -\tan(\theta)\n * \cot(-\theta) = -\cot(\theta)\n * \sec(-\theta) = \sec(\theta)\n * \csc(-\theta) = -\csc(\theta)\n * **Complementary Angle (90^\circ - \theta):**\n * \sin(90^\circ - \theta) = \cos(\theta)\n * \cos(90^\circ - \theta) = \sin(\theta)\n * \tan(90^\circ - \theta) = \cot(\theta)\n * \cot(90^\circ - \theta) = \tan(\theta)\n * \sec(90^\circ - \theta) = \csc(\theta)\n * \csc(90^\circ - \theta) = \sec(\theta)\n * **Allied Angle (90^\circ + \theta):**\n * \sin(90^\circ + \theta) = \cos(\theta)\n * \cos(90^\circ + \theta) = -\sin(\theta)\n * \tan(90^\circ + \theta) = -\cot(\theta)\n * \cot(90^\circ + \theta) = -\tan(\theta)\n * \sec(90^\circ + \theta) = -\csc(\theta)\n * \csc(90^\circ + \theta) = \sec(\theta)\n * **Supplementary Angle (180^\circ - \theta):**\n * \sin(180^\circ - \theta) = \sin(\theta)\n * \cos(180^\circ - \theta) = -\cos(\theta)\n * \tan(180^\circ - \theta) = -\tan(\theta)\n * \cot(180^\circ - \theta) = -\cot(\theta)\n * \sec(180^\circ - \theta) = -\sec(\theta)\n * \csc(180^\circ - \theta) = \csc(\theta)\n * **Allied Angle (180^\circ + \theta):**\n * \sin(180^\circ + \theta) = -\sin(\theta)\n * \cos(180^\circ + \theta) = -\cos(\theta)\n * \tan(180^\circ + \theta) = \tan(\theta)\n * \cot(180^\circ + \theta) = \cot(\theta)\n * \sec(180^\circ + \theta) = -\sec(\theta)\n * \csc(180^\circ + \theta) = -\csc(\theta)\n * **Allied Angle (270^\circ - \theta):**\n * \sin(270^\circ - \theta) = -\cos(\theta)\n * \cos(270^\circ - \theta) = -\sin(\theta)\n * \tan(270^\circ - \theta) = \cot(\theta)\n * \cot(270^\circ - \theta) = \tan(\theta)\n * \sec(270^\circ - \theta) = -\csc(\theta)\n * \csc(270^\circ - \theta) = -\sec(\theta)\n * **Allied Angle (270^\circ + \theta):**\n * \sin(270^\circ + \theta) = -\cos(\theta)\n * \cos(270^\circ + \theta) = \sin(\theta)\n * \tan(270^\circ + \theta) = -\cot(\theta)\n * \cot(270^\circ + \theta) = -\tan(\theta)\n * \sec(270^\circ + \theta) = \csc(\theta)\n * \csc(270^\circ + \theta) = -\sec(\theta)\n * **Allied Angle (360^\circ - \theta):**\n * \sin(360^\circ - \theta) = -\sin(\theta)\n * \cos(360^\circ - \theta) = \cos(\theta)\n * \tan(360^\circ - \theta) = -\tan(\theta)\n * \cot(360^\circ - \theta) = -\cot(\theta)\n * \sec(360^\circ - \theta) = \sec(\theta)\n * \csc(360^\circ - \theta) = -\csc(\theta)\n * **Periodic Angle (360^\circ + \theta):**\n * \sin(360^\circ + \theta) = \sin(\theta)\n * \cos(360^\circ + \theta) = \cos(\theta)\n * \tan(360^\circ + \theta) = \tan(\theta)\n * \cot(360^\circ + \theta) = \cot(\theta)\n * \sec(360^\circ + \theta) = \sec(\theta)\n * \csc(360^\circ + \theta) = \csc(\theta)\n\n# Quadrant Signs and the ASTC Rule\n\n![Diagram of the four Cartesian quadrants showing positive trigonometric ratios in each quadrant](https://assets.knowt.com/pdf-flow-prod/1b11cb22-f682-49ee-8c79-6c80e0a55a62-figures/12.png)\n\n* **Quadrant Breakdown:**\n * **Quadrant I:**\n * Angle range: 0^\circ \text{ to } 90^\circ\n * Positive ratios: All trigonometric functions are positive\n * Memory cue: All positive\n * **Quadrant II:**\n * Angle range: 90^\circ \text{ to } 180^\circ\n * Positive ratios: \sin, \csc\n * Memory cue: Sine positive\n * **Quadrant III:**\n * Angle range: 180^\circ \text{ to } 270^\circ\n * Positive ratios: \tan, \cot\n * Memory cue: Tangent positive\n * **Quadrant IV:**\n * Angle range: 270^\circ \text{ to } 360^\circ\n * Positive ratios: \cos, \sec\n * Memory cue: Cosine positive\n* **ASTC Memory Mnemonic:**\n * Mnemonic phrase: **Add Sugar To Coffee**\n * Letter assignments across quadrants:\n * **A** — **Add:** Quadrant I, all positive\n * **S** — **Sugar:** Quadrant II, \sin, \csc positive\n * **T** — **To:** Quadrant III, \tan, \cot positive\n * **C** — **Coffee:** Quadrant IV, \cos, \sec positive\n * Schematic progression: Add Sugar To Coffee \implies I: A, II: S, III: T, IV: C\n\n# Standard Angles and Exact Values\n\n* **Exact Values by Angle:**\n * **Angle 0^\circ:**\n * \sin(0^\circ) = 0\n * \cos(0^\circ) = 1\n * \tan(0^\circ) = 0\n * \cot(0^\circ) = \text{undefined}\n * \sec(0^\circ) = 1\n * \csc(0^\circ) = \text{undefined}\n * **Angle 30^\circ:**\n * \sin(30^\circ) = \frac{1}{2}\n * \cos(30^\circ) = \frac{\sqrt{3}}{2}\n * \tan(30^\circ) = \frac{1}{\sqrt{3}}\n * \cot(30^\circ) = \sqrt{3}\n * \sec(30^\circ) = \frac{2}{\sqrt{3}}\n * \csc(30^\circ) = 2\n * **Angle 45^\circ:**\n * \sin(45^\circ) = \frac{\sqrt{2}}{2}\n * \cos(45^\circ) = \frac{\sqrt{2}}{2}\n * \tan(45^\circ) = 1\n * \cot(45^\circ) = 1\n * \sec(45^\circ) = \sqrt{2}\n * \csc(45^\circ) = \sqrt{2}\n * **Angle 60^\circ:**\n * \sin(60^\circ) = \frac{\sqrt{3}}{2}\n * \cos(60^\circ) = \frac{1}{2}\n * \tan(60^\circ) = \sqrt{3}\n * \cot(60^\circ) = \frac{1}{\sqrt{3}}\n * \sec(60^\circ) = 2\n * \csc(60^\circ) = \frac{2}{\sqrt{3}}\n * **Angle 90^\circ:**\n * \sin(90^\circ) = 1\n * \cos(90^\circ) = 0\n * \tan(90^\circ) = \text{undefined}\n * \cot(90^\circ) = 0\n * \sec(90^\circ) = \text{undefined}\n * \csc(90^\circ) = 1\n* **Fast Construction Pattern Algorithm:**\n * Sine sequence generation for angles 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ:\n * \sin(\theta) = \frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\n * Cosine sequence: Reverse the exact generated sequence of sine values.\n * Remaining four ratios:\n * \tan = \sin / \cos\n * \cot = \cos / \sin\n * \sec = 1 / \cos\n * \csc = 1 / \sin\n\n# Double-Angle, Half-Angle, and Triple-Angle Identities\n\n* **Double-Angle Identities:**\n * **Sine:**\n * \sin(2\theta) = 2\sin(\theta)\cos(\theta)\n * **Cosine (Three Interchangeable Forms):**\n * Standard difference of squares: \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\n * Cosine-only form: \cos(2\theta) = 2\cos^2(\theta) - 1\n * Sine-only form: \cos(2\theta) = 1 - 2\sin^2(\theta)\n * **Tangent:**\n * \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\n* **Half-Angle Identities:**\n * **Cosine and Sine Squared Forms:**\n * \cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2}\n * \sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{2}\n * **Cosine and Sine Radical Forms:**\n * \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}\n * \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}\n * Sign specification: The \pmsignischosenbasedentirelyuponthequadrantinwhichthehalf−anglesign is chosen based entirely upon the quadrant in which the half-angle\frac{\theta}{2} resides.\n * **Tangent Half-Angle Forms:**\n * Squared form: \tan^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{1 + \cos(\theta)}\n * Radical form: \tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}\n * Rationalized alternative quotient forms: \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)}\n* **Triple-Angle Identities:**\n * **Sine:**\n * \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\n * **Cosine:**\n * \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)\n * **Tangent:**\n * \tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}\n\n# Product-to-Sum and Sum-to-Product Transformations\n\n* **Product-to-Sum Identities:**\n * Product of sine and cosine: 2\sin(\alpha)\cos(\beta) = \sin(\alpha + \beta) + \sin(\alpha - \beta)\n * Product of cosine and sine: 2\cos(\alpha)\sin(\beta) = \sin(\alpha + \beta) - \sin(\alpha - \beta)\n * Product of two cosines: 2\cos(\alpha)\cos(\beta) = \cos(\alpha + \beta) + \cos(\alpha - \beta)\n * Product of two sines: 2\sin(\alpha)\sin(\beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)\n* **Sum-to-Product Identities:**\n * Sum of sines: \sin(P) + \sin(Q) = 2\sin\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)\n * Difference of sines: \sin(P) - \sin(Q) = 2\cos\left(\frac{P + Q}{2}\right)\sin\left(\frac{P - Q}{2}\right)\n * Sum of cosines: \cos(P) + \cos(Q) = 2\cos\left(\frac{P + Q}{2}\right)\cos\left(\frac{P - Q}{2}\right)\n * Difference of cosines: \cos(P) - \cos(Q) = -2\sin\left(\frac{P + Q}{2}\right)\sin\left(\frac{P - Q}{2}\right)\n* **Structural Duality:** Product-to-sum and sum-to-product transformations represent reverse mathematical families.\n\n# Sine and Cosine Algebraic Forms and Distance Formula Motivation\n\n* **Common Special Expressions in Sine and Cosine Terms:**\n * Linear ratios:\n * \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\n * \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\n * \sec(\theta) = \frac{1}{\cos(\theta)}\n * \csc(\theta) = \frac{1}{\sin(\theta)}\n * Squared ratios:\n * \tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}\n * \cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}\n * \sec^2(\theta) = \frac{1}{\cos^2(\theta)}\n * \csc^2(\theta) = \frac{1}{\sin^2(\theta)}\n * Pythagorean squared expressions:\n * 1 + \tan^2(\theta) = \frac{1}{\cos^2(\theta)} = \sec^2(\theta)\n * 1 + \cot^2(\theta) = \frac{1}{\sin^2(\theta)} = \csc^2(\theta)\n* **Distance Formula Geometric Connection:**\n * General Cartesian distance formula between points P(x_1, y_1)andandQ(x_2, y_2):\n * PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n * Theoretical motivation: This distance formula is evaluated with coordinates formulated via sine and cosine on the Cartesian plane to motivate and establish the fundamental trigonometric law.\n\n# Strategic Exam Checklist and Memory Map\n\n* **Core Operational Checklist:**\n * Memorize all six trigonometric ratios and their corresponding reciprocals.\n * Recall \sin^2 + \cos^2 = 1 immediately without hesitation.\n * Master both quotient identities (\tanandand\cot).\n * Commit all signs for \sin(\alpha \pm \beta)andand\cos(\alpha \pm \beta) to memory.\n * For \tan(\alpha \pm \beta), actively track that the algebraic sign flips in the denominator.\n * For allied angles, execute a two-step decision process:\n * First, determine if the function retains the same ratio or switches to its co-ratio based on odd vs. even multiples of 90^\circ.\n * Second, evaluate the final sign (+\text{ or } -) according to the termination quadrant.\n * Recognize that double-angle cosine exhibits three interchangeable forms suited for targeted cancellation.\n * Account for quadrant location when selecting the positive or negative radical branch for half-angle evaluations.\n * Recognize that product-to-sum and sum-to-product formulas belong to mutually inverse algebraic families.\n * Ensure a variable divisor cannot equal zero without verifying domain restrictions.\n* **One-Line Memory Map:**\n * \text{Ratios} \rightarrow \text{Pythagorean} \rightarrow \text{Compound angles} \rightarrow \text{Allied angles} \rightarrow \text{Double} \rightarrow \text{Half} \rightarrow \text{Triple} \rightarrow \text{Product/Sum transformations}$$