Using Fundamental Identities Study Notes

Chapter 5.1 - Using Fundamental Identities

Fundamental Trigonometric Identities

Fundamental Trigonometric Identities are equations that are true for all angles. They serve as foundational tools in trigonometry for simplifying expressions and solving equations.

Reciprocal Identities

The Reciprocal Identities express trigonometric functions in terms of their reciprocals:

  • extsinheta=1cscθext{sin} heta = \frac{1}{\text{csc} \theta}
  • extcosθ=1secθext{cos} \theta = \frac{1}{\text{sec} \theta}
  • exttanθ=1cotθext{tan} \theta = \frac{1}{\text{cot} \theta}
  • cscθ=1sinθ\text{csc} \theta = \frac{1}{\text{sin} \theta}
  • secθ=1cosθ\text{sec} \theta = \frac{1}{\text{cos} \theta}
  • cotθ=1tanθ\text{cot} \theta = \frac{1}{\text{tan} \theta}
Quotient Identities

The Quotient Identities define the tangent and cotangent in terms of sine and cosine:

  • tanθ=sinθcosθ\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}
  • cotθ=cosθsinθ\text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta}
Pythagorean Identities

The Pythagorean Identities arise from the Pythagorean theorem:

  • sin2θ+cos2θ=1\text{sin}^2 \theta + \text{cos}^2 \theta = 1
  • tan2θ+1=sec2θ\text{tan}^2 \theta + 1 = \text{sec}^2 \theta
  • 1+cot2θ=csc2θ1 + \text{cot}^2 \theta = \text{csc}^2 \theta
Cofunction Identities

Cofunction Identities relate sine and cosine, tangent and cotangent, etc., for complementary angles:

  • sin(π2−θ)=cosθ\text{sin}\left(\frac{\pi}{2} - \theta\right) = \text{cos} \theta
  • tan(π2−θ)=cotθ\text{tan}\left(\frac{\pi}{2} - \theta\right) = \text{cot} \theta
Odd-Even Identities

Odd-even identities describe how the values of functions behave with negative inputs:

  • sin(−θ)=−sinθ\text{sin}(-\theta) = -\text{sin} \theta
  • csc(−θ)=−cscθ\text{csc}(-\theta) = -\text{csc} \theta
  • cos(−θ)=cosθ\text{cos}(-\theta) = \text{cos} \theta
  • sec(−θ)=secθ\text{sec}(-\theta) = \text{sec} \theta
  • tan(−θ)=−tanθ\text{tan}(-\theta) = -\text{tan} \theta
  • cot(−θ)=−cotθ\text{cot}(-\theta) = -\text{cot} \theta

Example Problems

Example 1: Determine cos x
  • Given: sinx=513\text{sin} x = \frac{5}{13} and \text{tan} x > 0
  • Find: cosx\text{cos} x
  • Using the Pythagorean Identity:
    • sin2x+cos2x=1\text{sin}^2 x + \text{cos}^2 x = 1
    • (513)2+cos2x=1\left(\frac{5}{13}\right)^2 + \text{cos}^2 x = 1
  • Calculate:
    • 25169+cos2x=1\frac{25}{169} + \text{cos}^2 x = 1
    • cos2x=1−25169=144169\text{cos}^2 x = 1 - \frac{25}{169} = \frac{144}{169}
  • Thus:
    • cosx=1213\text{cos} x = \frac{12}{13} (positive since \text{tan} x > 0).
Example 2: Simplify the expression ( \text{sin} x \cdot \text{cot} x )
  • Approach: Use the quotient identity:
    • sinx⋅cotx=sinx⋅cosxsinx\text{sin} x \cdot \text{cot} x = \text{sin} x \cdot \frac{\text{cos} x}{\text{sin} x}
  • Result: Cancelling sinx\text{sin} x gives:
    • cosx\text{cos} x
Additional Simplifications
a) Simplification of cos2x⋅cscx−cscx\text{cos}^2 x \cdot \text{csc} x - \text{csc} x
  • Factor Out: cscx(cos2x−1)\text{csc} x (\text{cos}^2 x - 1)
  • Apply: sin2x+cos2x=1\text{sin}^2 x + \text{cos}^2 x = 1 (Pythagorean identity).
f) Simplification of 4tan⁡2x+tan⁡x−34 \tan^2 x + \tan x - 3
  • Rearrangement: tan⁡2x+tan⁡x−3=0\tan^2 x + \tan x - 3 = 0
  • Factoring the expression: tan⁡x(tan⁡x+1)\tan x (\tan x + 1) gives:
    • 4tan⁡x=34 \tan x = 3.
  • Resolve: tan⁡x=34\tan x = \frac{3}{4}.

Homework

  • Textbook pages 355 - 356, problems 21 - 31 (odd), 35 - 49 (odd), 61.
  • Make sure to apply the various identities learned.