Comprehensive Trigonometry Study Guide: Unit Circle, Right Triangle Trigonometry, and Fundamental Identities

Unit Circle Foundations: Inputs, Outputs, and Function Values

  • Domain and Range of Trigonometric Functions:

    • Output Values (Range): The outputs for sine and cosine are constrained between −1-1 and 11 (i.e., the interval [−1,1][-1, 1]).

    • Input Values (Domain): The domain consists of all real numbers ((−∞,∞)(-\infty, \infty)). While the unit circle explicitly depicts values from 00 to 2π2\pi, inputs can extend infinitely in either direction (e.g., 4π4\pi, 6π6\pi, 8π8\pi, or negative angle values).

  • Unit Circle Coordinate Correspondence:

    • Cosine: Corresponds directly to the xx-coordinate on the unit circle: cos⁡(t)=x\cos(t) = x.

    • Sine: Corresponds directly to the yy-coordinate on the unit circle: sin⁡(t)=y\sin(t) = y.

  • Coordinate Evaluation Examples:

    • Evaluation at π4\frac{\pi}{4}:

    • cos⁡(π4)=22\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

    • Evaluation at −π4-\frac{\pi}{4}:

    • Moving an equal distance in the negative (clockwise) direction lands on the coordinate at 7π4\frac{7\pi}{4}.

    • cos⁡(−π4)=22\cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

    • Evaluation at π2\frac{\pi}{2} and −π2-\frac{\pi}{2}:

    • cos⁡(π2)=0\cos\left(\frac{\pi}{2}\right) = 0

    • cos⁡(−π2)=0\cos\left(-\frac{\pi}{2}\right) = 0

Even and Odd Trigonometric Functions

  • Even Functions:

    • Definition & Symmetry: A function is even if evaluating it at −t-t produces the same result as evaluating it at tt. Cosine is symmetric with respect to the xx-axis.

    • Even Functions List:

    • Cosine: cos⁡(−t)=cos⁡(t)\cos(-t) = \cos(t)

    • Secant: sec⁡(−t)=sec⁡(t)\sec(-t) = \sec(t)

    • Reciprocal Relationship: Secant is the reciprocal of cosine (i.e., sec⁡(t)=1cos⁡(t)\sec(t) = \frac{1}{\cos(t)}), making its even behavior identical to cosine.

  • Odd Functions:

    • Evaluations:

    • sin⁡(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

    • sin⁡(−π6)=−12\sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}

    • sin⁡(π4)=22\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

    • sin⁡(−π4)=−22\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}

    • sin⁡(−π3)=−32\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}

    • Definition & Symmetry: A function is odd if f(−t)=−f(t)f(-t) = -f(t). These functions exhibit symmetry with respect to y=xy = x

    • Odd Functions List:

    • Sine: sin⁡(−t)=−sin⁡(t)\sin(-t) = -\sin(t)

    • Cosecant: csc⁡(−t)=−csc⁡(t)\csc(-t) = -\csc(t)

    • Tangent: tan⁡(−t)=−tan⁡(t)\tan(-t) = -\tan(t)

    • Cotangent: cot⁡(−t)=−cot⁡(t)\cot(-t) = -\cot(t)

  • Tangent and Cotangent on the Unit Circle:

    • Tangent is not explicitly listed as a single coordinate on the unit circle; it is computed by dividing the yy-coordinate by the xx-coordinate:     tan⁡(t)=yx\tan(t) = \frac{y}{x}

    • Cotangent is computed by dividing the xx-coordinate by the yy-coordinate:     cot⁡(t)=xy\cot(t) = \frac{x}{y}

Trigonometric Periodicity and Problem Solving

  • Concept of Periodicity:

    • Adding or subtracting integer multiples of the period 2π2\pi to any angle lands at the exact same point on the unit circle.

  • Evaluating Functions Using Periodicity:

    • Problem 1: Evaluate sin⁡(13π6)\sin\left(\frac{13\pi}{6}\right).

    • Separate 13π6\frac{13\pi}{6} into a multiple of 2π2\pi plus a remaining angle:       13π6=2π+π6=12π6+π6\frac{13\pi}{6} = 2\pi + \frac{\pi}{6} = \frac{12\pi}{6} + \frac{\pi}{6}

    • Because adding 2π2\pi returns to the same position:       sin⁡(13π6)=sin⁡(π6)=12\sin\left(\frac{13\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

    • Problem 2: Evaluate cos⁡(−7π2)\cos\left(-\frac{7\pi}{2}\right).

    • Method A (Even Property): cos⁡(−7π2)=cos⁡(7π2)\cos\left(-\frac{7\pi}{2}\right) = \cos\left(\frac{7\pi}{2}\right).

    • Method B (Using Period): One period equal to 2π2\pi is 4π2\frac{4\pi}{2}. Express cos⁡(−7π2)\cos\left(-\frac{7\pi}{2}\right) relative to multiples of 4π2\frac{4\pi}{2}, such as π2−8π2\frac{\pi}{2} - \frac{8\pi}{2}.

    • Problem 3: Given sin⁡(t)=45\sin(t) = \frac{4}{5}, find sin⁡(−t)\sin(-t).

    • Apply the odd function identity sin⁡(−t)=−sin⁡(t)\sin(-t) = -\sin(t).

    • Substitute the given value:       sin⁡(−t)=−45\sin(-t) = -\frac{4}{5}

Course Logistics, Homework, and Exam Coverage

  • Homework Assignment:

    • Section 1.2: Exercises 55 through 4747 odd (5−475-47 odd).

    • Homework is not collected or graded, but completing it is the best method to prepare for exams.

  • Exam 1 Details:

    • Scope: Covers sections 1.1, 1.2, 1.3, and 1.4.

Right Triangle Trigonometry

  • Applicability:

    • Right triangle trigonometry applies exclusively to triangles containing one 90∘90^\circ angle.

    • Non-right triangles (where no angle equals 90∘90^\circ) require the Law of Sines or Law of Cosines.

  • Triangle Terminology relative to Acute Angle θ\theta:

    • Hypotenuse: The side directly opposite the 90∘90^\circ right angle.

    • Opposite: The side directly across from the angle θ\theta.

    • Adjacent: The side immediately next to the angle θ\theta.

  • Definitions of the Six Trigonometric Functions:

    • sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

    • cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

    • tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

    • csc⁡(θ)=hypotenuseopposite\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}

    • sec⁡(θ)=hypotenuseadjacent\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}

    • cot⁡(θ)=adjacentopposite\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}

  • Example: Solving a Right Triangle:

    • Given: Opposite side = 44, Adjacent side = 33.

    • Step 1: Find the hypotenuse (cc) using the Pythagorean Theorem:     a2+b2=c2a^2 + b^2 = c^2     32+42=c23^2 + 4^2 = c^2     9+16=c29 + 16 = c^2     25=c225 = c^2     25=c  ⟹  c=5\sqrt{25} = c \implies c = 5

    • Note on Signs: Solving c2=25c^2 = 25 by introducing a square root mathematically yields ±5\pm 5. However, because geometric distance without defined orientation/direction must be positive, c=5c = 5.

    • This forms a standard 3−4−53-4-5 right triangle (and applies to all scalar multiples of 3−4−53-4-5).

    • Step 2: Calculate the Six Trigonometric Ratios:

    • sin⁡(θ)=45\sin(\theta) = \frac{4}{5}

    • cos⁡(θ)=35\cos(\theta) = \frac{3}{5}

    • tan⁡(θ)=43\tan(\theta) = \frac{4}{3}

    • csc⁡(θ)=54\csc(\theta) = \frac{5}{4}

    • sec⁡(θ)=53\sec(\theta) = \frac{5}{3}

    • cot⁡(θ)=34\cot(\theta) = \frac{3}{4}

Special Angles and Trigonometric Ratios

  • Origin:

    • Special angle values are derived from two special right triangles embedded within the unit circle: the 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle and the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle.

  • Summary Table of Special Angle Ratios:

    • 30∘30^\circ (π6\frac{\pi}{6} radians):

    • sin⁡(30∘)=sin⁡(π6)=12\sin\left(30^\circ\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

    • cos⁡(30∘)=cos⁡(π6)=32\cos\left(30^\circ\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}

    • tan⁡(30∘)=tan⁡(π6)=1232=13=33\tan\left(30^\circ\right) = \tan\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

    • 45∘45^\circ (π4\frac{\pi}{4} radians):

    • sin⁡(45∘)=sin⁡(π4)=22\sin\left(45^\circ\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

    • cos⁡(45∘)=cos⁡(π4)=22\cos\left(45^\circ\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

    • tan⁡(45∘)=tan⁡(π4)=2222=1\tan\left(45^\circ\right) = \tan\left(\frac{\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1

    • 60∘60^\circ (π3\frac{\pi}{3} radians):

    • sin⁡(60∘)=sin⁡(π3)=32\sin\left(60^\circ\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}

    • cos⁡(60∘)=cos⁡(π3)=12\cos\left(60^\circ\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}

    • tan⁡(60∘)=tan⁡(π3)=3212=3\tan\left(60^\circ\right) = \tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}

Complementary Angles and Cofunctions

  • Definition of Complementary Angles:

    • Two acute angles are complementary if their sum equals 90∘90^\circ (or π2\frac{\pi}{2} radians).

  • Cofunction Rule:

    • Cofunctions of complementary angles are equal.

    • Demonstrations:

    • sin⁡(30∘)=12\sin\left(30^\circ\right) = \frac{1}{2} and cos⁡(60∘)=12\cos\left(60^\circ\right) = \frac{1}{2} (since 30∘+60∘=90∘30^\circ + 60^\circ = 90^\circ).

    • sin⁡(60∘)=32\sin\left(60^\circ\right) = \frac{\sqrt{3}}{2} and cos⁡(30∘)=32\cos\left(30^\circ\right) = \frac{\sqrt{3}}{2}.

    • sin⁡(45∘)=cos⁡(45∘)=22\sin\left(45^\circ\right) = \cos\left(45^\circ\right) = \frac{\sqrt{2}}{2}.

    • If sin⁡(15∘)=a\sin\left(15^\circ\right) = a, then cos⁡(75∘)=a\cos\left(75^\circ\right) = a because 15∘+75∘=90∘15^\circ + 75^\circ = 90^\circ.

Fundamental Trigonometric Identities

  • Cofunction Identities:

    • sin⁡(π2−θ)=cos⁡(θ)\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)

    • cos⁡(π2−θ)=sin⁡(θ)\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)

    • Similar cofunction relationships exist between tangent and cotangent, as well as secant and cosecant.

  • Reciprocal Identities:

    • sin⁡(θ)=1csc⁡(θ)\sin(\theta) = \frac{1}{\csc(\theta)} and csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

    • cos⁡(θ)=1sec⁡(θ)\cos(\theta) = \frac{1}{\sec(\theta)} and sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}

    • tan⁡(θ)=1cot⁡(θ)\tan(\theta) = \frac{1}{\cot(\theta)} and cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}

  • Quotient Identities:

    • 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)}

  • Pythagorean Identities:

    • sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

    • 1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

    • 1+cot⁡2(θ)=csc⁡2(θ)1 + \cot^2(\theta) = \csc^2(\theta)

  • Algebraic Manipulations of Pythagorean Identities:

    • cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1

    • 1−sin⁡2(θ)=cos⁡2(θ)1 - \sin^2(\theta) = \cos^2(\theta)

    • 1−cos⁡2(θ)=sin⁡2(θ)1 - \cos^2(\theta) = \sin^2(\theta)

    • sec⁡2(θ)−1=tan⁡2(θ)\sec^2(\theta) - 1 = \tan^2(\theta)

    • 1−sec⁡2(θ)=−tan⁡2(θ)1 - \sec^2(\theta) = -\tan^2(\theta)

Formula Sheet Access and Reference Materials

  • Locating the Reference Sheet in the E-Book:

    • Navigate to the formula cards tab at location A 26 / ES 4 in the e-book.

  • Exam Reference Sheet Scope:

    • The provided formula sheet for exams spans from page ES 4 down to immediately before the section titled "Formulas from Geometry".

    • It includes the full unit circle diagram.