Chapter 10: Trig Identities and Proving Identities

Graphs and Transformations

  • The discussion centers on recognizing a graph that most closely resembles a tangent-shaped graph.
  • Goal: identify a formula in terms of tangent, and note that shifting it left or right complicates things.
  • Vertical asymptotes:
    • Tangent does not have a vertical asymptote at zero and essentially straddles the y-axis.
    • Cotangent is the other standard trig function with vertical asymptotes (at multiples of
      x=k πx = k\,\pi).
  • The graph in question is not exactly cotangent; it’s flipped relative to cotangent.
    • A simple fix is to apply a minus sign in front (i.e., multiply by
      −1-1), which is often easier than horizontal shifts.
  • Upside-down form: use a negative out front to reflect the graph.
  • Period considerations:
    • The normal period of tangent is π\pi.
    • The graph being discussed appears to have a period of 2π2\pi until a stretch is determined.
    • The speaker leaves the period blanks for now, indicating uncertainty about stretch and exact period.
  • Vertical/vertical-logical shift: the graph needs to be shifted up by 1 unit (e.g., from starting points like (-1, 0, 1) to (0, 1, 2) in the relevant y-values).
  • Summary takeaway: identify a tangent-like transformation, use a negative sign if flipped, and consider vertical shifts before fixing the period.

Chapter 10: Identities and Proving Identities

  • Chapter focus: developing methods to prove identities and use these identities to evaluate expressions not on the unit circle (e.g., 15° values).
  • Foundational practice: memorize the identities already discussed; they will appear on the formula sheet.
  • Core guidance: you should know these identities by heart and be able to recall them quickly today.
  • Important caveat: there will be additional identities provided on the formula sheet, but memorization remains crucial for quick recall during problem solving.

What an identity is

  • An identity is an equation that is true for every value in the domain (i.e., for all angles).
  • Notation: angles can be represented with different symbols (a, b, θ, x, etc.), but the identity must hold for all of them.
  • Distinction:
    • Prove an identity: show the equality holds for all valid angles.
    • Verify an identity: if the problem states 'verify', you may treat the identity as true and show the left-hand side equals the right-hand side.
  • Strategy emphasis: start with the side that looks messier; manipulate algebraically to reveal equivalence.
  • Example idea: use algebraic moves (multiplication/division by same nonzero quantity, factoring, expanding) to reveal equality, while ensuring you can justify each step.

Key trig identities (recap and use in proofs)

  • Reciprocal identities:
    • Tangent and cotangent relationship:
      cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}
    • (Note: other reciprocal identities for sine/cosine include csc x = 1/ sin x and sec x = 1/ cos x, though only cot x is explicitly mentioned here.)
  • Pythagorean identities:
    • Basic:
      sin⁡2x+cos⁡2x=1\sin^{2}x + \cos^{2}x = 1
    • Derived:
      tan⁡2x+1=sec⁡2x\tan^{2}x + 1 = \sec^{2}x
    • Derivation hint: divide the left-hand side of the basic identity by cos⁡2x\cos^{2}x to obtain the derived form.
  • Using signs in quadrants (sign conventions): some identities depend on the quadrant, e.g., signs of sine/c cosine; ensure the sign is consistent when substituting.
  • Expressing sine in terms of cosine (and vice versa):
    • From the basic identity:
      sin⁡2x=1−cos⁡2x\sin^{2}x = 1 - \cos^{2}x
    • Consequently,
      sin⁡x=±1−cos⁡2x\sin x = \pm \sqrt{1 - \cos^{2}x}
    • It's common to rewrite expressions entirely in terms of sine and cosine before simplifying.
  • Practical note: when reducing complex expressions, replacing secant with its cosine form can simplify handling of quotients.

Proving identities: a practical workflow

  • Step 1: Examine the left-hand side (LHS) and right-hand side (RHS) of the identity.
  • Step 2: Look for a starting move that is obviously equivalent (e.g., replace secant with 1/ cos, or rewrite tangent as sine over cosine).
  • Step 3: Simplify consistently, keeping track of how each operation transforms the expression.
  • Step 4: If you get a messy quotient, consider rewriting everything in terms of sine and cosine and/or factor where useful.
  • Step 5: If a path stalls, back up and try a different starting substitution. The key is to begin with something you know is equivalent and then proceed step by step so others can follow your logic.
  • Step 6: The end goal is to show the left side is equal to the right side by a clear chain of equal expressions, with each step justified.

Worked approach to a sample identity (LHS: sec x sin x + cos x)

  • Start with:
    LHS=sec⁡xsin⁡x+cos⁡x\text{LHS} = \sec x \sin x + \cos x
  • Replace secant:
    LHS=sin⁡xcos⁡x+cos⁡x\text{LHS} = \frac{\sin x}{\cos x} + \cos x
  • Get a common denominator:
    LHS=sin⁡xcos⁡x+cos⁡2xcos⁡x=sin⁡x+cos⁡2xcos⁡x\text{LHS} = \frac{\sin x}{\cos x} + \frac{\cos^{2}x}{\cos x} = \frac{\sin x + \cos^{2}x}{\cos x}
  • Express cos^2 x using the Pythagorean identity:
    cos⁡2x=1−sin⁡2x\cos^{2}x = 1 - \sin^{2}x
  • Substitute:
    LHS=sin⁡x+(1−sin⁡2x)cos⁡x=1+sin⁡x−sin⁡2xcos⁡x\text{LHS} = \frac{\sin x + (1 - \sin^{2}x)}{\cos x} = \frac{1 + \sin x - \sin^{2}x}{\cos x}
  • (Notes on further simplification): continuing from here would require either expressing everything in terms of sine or cosine, or using other identities to simplify the numerator and/or denominator. The key is to proceed step by step so the path is transparent to someone else following your work.
  • Important practice point: do not skip steps; the goal is a transparent, reproducible chain of equalities.

Algebraic techniques relevant to trigonometric identities

  • Factoring techniques (refer to algebra basics) that often appear in identity proofs:
    • Difference of squares:
      a2−b2=(a−b)(a+b)a^{2} - b^{2} = (a - b)(a + b)
    • Sum of cubes:
      a3+b3=(a+b)(a2−ab+b2)a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})
    • Difference of cubes:
      a3−b3=(a−b)(a2+ab+b2)a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
  • When a problem asks you to prove an identity, you may need to manipulate each side separately before showing they are equal; however, you should maintain the invariant that you are not introducing extraneous steps that would not hold for all angles.
  • Strategy: start with a form you know is equivalent, then simplify until you reach a common expression or a form that clearly matches the other side.

Problem-solving strategies and learner tips

  • Start with something you know is equivalent; if it doesn’t help, backtrack and try another route.
  • The objective is to replace expressions with equivalent expressions and to show the chain of equivalences clearly to the reader.
  • Remember: there can be multiple valid paths to the same identity; the key is to be transparent about each step.
  • In exam settings, be prepared to show a complete, step-by-step derivation so a grader can follow your reasoning.

Quick algebra refresher you should know by heart

  • Factor special quadratics (from algebra):
    • Difference of squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
    • Sum of cubes: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
    • Difference of cubes: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
  • Practice focus: recognize these patterns to simplify trigonometric expressions efficiently.

Off-topic note (digressions in the transcript)

  • A segment of the transcript veered into casual conversations about video games (e.g., Ark Survival Ascended, Ark: Survival Evolved) and personal gaming discussions.
  • These digressions are not part of the math content and should be ignored for study purposes.
  • Some numeric chatter about hours played and platform specifics appeared; not relevant to the identities material.

Connections to broader topics and real-world relevance

  • Identities underpin how we transform and simplify expressions in signal processing, physics, engineering, and computer graphics.
  • Understanding period, symmetry, and shifts in trig graphs translates to waveform analysis and rotational motion problems.
  • Mastery of algebraic factoring improves problem-solving speed across calculus (limits, derivatives) and linear algebra (eigenvalue proofs, transforms).

Quick reference formulas (recap)

  • Reciprocal relation:
    cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}
  • Pythagorean identities:
    sin⁡2x+cos⁡2x=1 ,\sin^{2}x + \cos^{2}x = 1\,,
    tan⁡2x+1=sec⁡2x .\tan^{2}x + 1 = \sec^{2}x\,.
  • Basic sine-cosine identity for manipulation:
    sin⁡2x=1−cos⁡2x ,\sin^{2}x = 1 - \cos^{2}x\,,
    cos⁡2x=1−sin⁡2x .\cos^{2}x = 1 - \sin^{2}x\,.
  • Tangent and secant quotients:
    tan⁡x=sin⁡xcos⁡x,sec⁡x=1cos⁡x,sec⁡2x=1cos⁡2x .\tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x}, \quad \sec^{2}x = \frac{1}{\cos^{2}x}\,.
  • Periods (for reference):
    • period(tan⁡x)=π\text{period}(\tan x) = \pi
    • For a transformed or stretched graph, the period may be 2π2\pi or another value depending on the transformation applied.
  • Typical transformation forms to recognize: shift, reflection, and dilation: e.g., tan⁡(x−h)\tan(x - h) (shift right by h), −tan⁡x-\tan x (reflection about the x-axis).