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Precalculus: Trigonometric Functions and Applications

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Terms in this set (20)

  • Exact values of trigonometric functions for 45° (π/4)
    In a right isosceles triangle with legs 1, hypotenuse is \(\sqrt{2}\). Then sin 45° = cos 45° = \(\frac{1}{\sqrt{2}}\), tan 45° = 1.
  • Exact values of trigonometric functions for 30° and 60°
    In a 30°-60°-90° triangle with hypotenuse 2, the shorter leg is 1 and the longer leg is \(\sqrt{3}\). Then sin 30° = 1/2, cos 30° = \(\frac{\sqrt{3}}{2}\), tan 30° = \(\frac{1}{\sqrt{3}}\).
  • Using calculator to approximate trigonometric values
    Set calculator mode to degrees or radians as needed. Use functions like cos, sin, tan directly. Round answers to two decimal places.
  • Definition of coterminal angles
    Two angles are coterminal if they share the same terminal side. Coterminal angles differ by multiples of 360° or \(2\pi\).
  • Signs of trigonometric functions in quadrants
    In quadrant I all are positive; II: sin and csc positive; III: tan and cot positive; IV: cos and sec positive.
  • Reference angle definition
    The acute angle formed between the terminal side of an angle and the x-axis. Used to find exact trig values for any angle.
  • Finding trig functions from a point on terminal side
    Given point (a,b) on terminal side, radius \(r=\sqrt{a^2+b^2}\), then sin θ = b/r, cos θ = a/r, tan θ = b/a.
  • Exact trig values for quadrantal angles
    At 0°, 90°, 180°, 270°, sin, cos, tan take values 0, ±1 or undefined depending on coordinates of points on axes.
  • Steps to find trig values for any angle
    1. Find reference angle α. 2. Find trig value at α. 3. Adjust sign based on quadrant of θ.
  • Formula for area of rain gutter opening bent at angle θ
    Area A = 15 cos θ (5 + 5 tan θ), where 15 is sheet width and 5 is length bent up.
  • Maximum area angle for rain gutter problem
    The angle θ that maximizes the area A is 60°, giving the largest opening for water flow.
  • Using tangent to find width of river
    Width b = a tan θ, where a is distance walked perpendicular to river and θ is measured angle.
  • Using angle of elevation to find height of cloud
    Height h = b tan θ, where b is horizontal distance from detector to projector and θ is angle of elevation.
  • Finding height of statue using two angles of elevation
    Height of statue = height to top - height to base, each found by h = distance × tan(angle).
  • Definition of six trigonometric functions using coordinates
    sin θ = b/r, cos θ = a/r, tan θ = b/a, csc θ = r/b, sec θ = r/a, cot θ = a/b, where (a,b) is point on terminal side and r = distance from origin.
  • Coterminal angle formula
    Coterminal angles: θ + 360°k or θ + 2πk, where k is any integer.
  • Signs of trig functions mnemonic
    All Students Take Calculus: Quadrant I all positive, II sin positive, III tan positive, IV cos positive.
  • Reference angle for angles in different quadrants
    Quadrant II: α = 180° - θ; III: α = θ - 180°; IV: α = 360° - θ.
  • Exact trig values using reference angles
    Use reference angle α to find trig value, then apply sign based on quadrant of original angle θ.
  • Tangent function undefined when
    Tangent is undefined when cos θ = 0, i.e., at 90° and 270° (π/2 and 3π/2 radians).