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Precalculus: Trigonometric Functions and Applications

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Termini in questo insieme (19)

  • Definition of sine, cosine, and tangent in a right triangle

    sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent

  • Values of sides in a right triangle with sides 7, 24, 25

    Opposite side = 7, Adjacent side = 24, Hypotenuse = 25

  • Exact sine, cosine, and tangent values for 30°, 45°, and 60°

    30°: sin = 1/2, cos = \(\frac{\sqrt{3}}{2}\), tan = \(\frac{1}{\sqrt{3}}\)

    45°: sin = cos = \(\frac{\sqrt{2}}{2}\), tan = 1

    60°: sin = \(\frac{\sqrt{3}}{2}\), cos = 1/2, tan = \(\sqrt{3}\)

  • Cofunction identities for sine and cosine

    sin A = cos (90° - A) and cos A = sin (90° - A)

  • Definition of a reference angle

    A reference angle is the acute angle between the terminal side of an angle and the x-axis.

  • How to find the reference angle for an angle in degrees

    Subtract the nearest x-axis angle (0°, 180°, or 360°) to get an acute angle between 0° and 90°.

  • Reference angle for 218°

    218° - 180° = 38°

  • Reference angle for 1387°

    1387° - 3 × 360° = 307°, then 360° - 307° = 53°

  • Using reference angles to find trig values for 210°

    Reference angle is 30°. Use 30° trig values and adjust sign based on quadrant III.

  • Steps to find trig function values for any angle θ
    1. Find coterminal angle between 0° and 360°
    2. Find reference angle
    3. Find trig values for reference angle
    4. Adjust signs based on quadrant
  • Finding cos(–240°) using reference angles

    –240° coterminal with 120°, reference angle 60°, cos negative in quadrant II, so cos(–240°) is negative.

  • Finding tan 675° using reference angles

    675° coterminal with 315°, reference angle 45°, tan negative in quadrant IV, so tan 675° = –1.

  • Sign of sine and cosine in quadrant II

    Sine is positive, cosine is negative in quadrant II.

  • Sign of sine and cosine in quadrant III

    Both sine and cosine are negative in quadrant III.

  • Using calculator to find trig values

    Set calculator to degree or radian mode and use sin, cos, tan, or their inverses to find approximate values.

  • Using inverse sine to find angle for sin θ ≈ 0.9677

    θ ≈ sin⁻¹(0.9677) = 75.4°

  • Using inverse tangent to find angle for tan θ ≈ 0.25 in radians

    θ ≈ tan⁻¹(0.25) = 0.24498 radians

  • Finding all solutions for cos θ = –\(\frac{\sqrt{2}}{2}\) in [0°, 360°)

    θ = 135° (quadrant II) and 225° (quadrant III)

  • Finding two angles in [0, 2π) for cos θ ≈ 0.3624

    First θ ≈ 1.2 radians (quadrant I), second θ ≈ 2π – 1.2 ≈ 5.08 radians (quadrant IV)