Trigonometry Flashcards

The Unit Circle

  • Equation of the unit circle: x2+y2=1x^2 + y^2 = 1
  • Sine function: Input is θ\theta, output is the y-coordinate.
  • Cosine function: Input is θ\theta, output is the x-coordinate.

Reference Angles

  • Reference angle θ′\theta': Acute angle formed by the terminal side of θ\theta and the horizontal axis.

The Six Trigonometric Functions

  • Tangent function: Input is θ\theta, output is yx\frac{y}{x}.
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • Reciprocal Identities:
    • Cosecant: Reciprocal of sine.
    • Secant: Reciprocal of cosine.
    • Cotangent: Reciprocal of tangent.

Right Triangle Trigonometry

  • sin⁡θ=yr\sin \theta = \frac{y}{r}, cos⁡θ=xr\cos \theta = \frac{x}{r}, tan⁡θ=yx\tan \theta = \frac{y}{x} where r=x2+y2r = \sqrt{x^2 + y^2}
  • Using "opp," "adj," and "hyp" for a right triangle:
    • sin⁡θ=opphyp\sin \theta = \frac{opp}{hyp}, csc⁡θ=hypopp\csc \theta = \frac{hyp}{opp}
    • cos⁡θ=adjhyp\cos \theta = \frac{adj}{hyp}, sec⁡θ=hypadj\sec \theta = \frac{hyp}{adj}
    • tan⁡θ=oppadj\tan \theta = \frac{opp}{adj}, cot⁡θ=adjopp\cot \theta = \frac{adj}{opp}

Trigonometric Identities

  • Cofunctions of complementary angles are equal.
    • sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos \theta
    • tan⁡(90∘−θ)=cot⁡θ\tan(90^\circ - \theta) = \cot \theta
    • sec⁡(90∘−θ)=csc⁡θ\sec(90^\circ - \theta) = \csc \theta
  • Reciprocal Identities:
    • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}
  • Quotient Identities:
    • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}
  • Pythagorean Identities:
    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta
    • 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Trigonometric Functions of Real Numbers

  • Trigonometric functions can use real numbers as inputs (in radians).

Other Trig Function Properties

  • Sine function:
    • Domain: (−∞,∞)(-\infty, \infty), Range: [−1,1][-1, 1], Period: 2π2\pi
  • Cosine function:
    • Domain: (−∞,∞)(-\infty, \infty), Range: [−1,1][-1, 1], Period: 2π2\pi
  • Even/Odd Functions:
    • Even: cosine
    • Odd: sine, tangent
  • Periodic Function: f(t+c)=f(t)f(t + c) = f(t)