Skip to main content
Indietro

Precalculus: Trigonometric Functions and Applications

I pulsanti di controllo sono stati cambiati in modalità "navigazione".
1/19

Termini in questo insieme (19)

  • What are the six trigonometric functions?
    Sine, cosine, tangent, cotangent, secant, and cosecant.
  • How is the distance r defined for a point (x, y) on the terminal side of an angle θ?
    r = \(\sqrt{x^2 + y^2}\), the distance from the origin to the point.
  • How do you find the six trigonometric functions given a point (x, y) on the terminal side of θ?
    Use sin θ = y/r, cos θ = x/r, tan θ = y/x, csc θ = r/y, sec θ = r/x, and cot θ = x/y.
  • What is the significance of choosing any point on the terminal side of θ to find trig functions?
    The trig function values are the same for any point on the terminal side because corresponding sides of similar triangles are proportional.
  • What are the values of sin 90° and cos 90°?
    sin 90° = 1 and cos 90° = 0.
  • Which trigonometric functions are undefined when the terminal side lies along the y-axis?
    Tangent and secant are undefined.
  • Which trigonometric functions are undefined when the terminal side lies along the x-axis?
    Cotangent and cosecant are undefined.
  • What are the reciprocal identities for cosine and secant?
    sec θ = 1 / cos θ and cos θ = 1 / sec θ.
  • How do signs of trig functions vary by quadrant?
    In quadrant I, all six functions are positive. In quadrant II, sine and cosecant are positive; others are negative. Similar sign patterns apply for quadrants III and IV.
  • If sin θ > 0 and tan θ < 0, in which quadrant is θ located?
    Quadrant II.
  • If cos θ < 0 and sec θ < 0, in which quadrants is θ located?
    Quadrants II and III.
  • What is the Pythagorean identity derived from x² + y² = r² divided by r²?
    sin² θ + cos² θ = 1.
  • What is the Pythagorean identity derived from dividing x² + y² = r² by x²?
    1 + tan² θ = sec² θ.
  • What is the Pythagorean identity derived from dividing x² + y² = r² by y²?
    1 + cot² θ = csc² θ.
  • What are the quotient identities for tangent and cotangent?
    tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.
  • How do you find sin θ and cos θ if tan θ = 4/3 and θ is in quadrant III?
    Both sin θ and cos θ are negative in quadrant III; use the ratio to find values accordingly.
  • How can sec θ be expressed in terms of sin θ if θ is in quadrant IV?
    sec θ = \(\frac{1}{\sqrt{1 - sin^2 θ}\)}, choosing the positive root since sec θ > 0 in quadrant IV.
  • How is the grade of a road modeled by y = -0.06x interpreted?
    The road has a -6% grade, meaning it drops 6 feet for every 100 feet horizontally.
  • How do you calculate grade resistance for a 3000-pound car on a road with grade -6%?
    Grade resistance = weight × sin θ ≈ 3000 × (-0.06) = -180 pounds, indicating downhill force.