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Right Triangle and Any Angle Trigonometry - Precalculus

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Termini in questo insieme (20)

  • What is the relationship between sine and cosine of complementary angles?

    sin θ = cos (90° − θ) and cos θ = sin (90° − θ). They are cofunctions of complementary angles.

  • What is the cofunction identity for tangent and cotangent?

    tan θ = cot (90° − θ) and cot θ = tan (90° − θ).

  • What is the cofunction identity for secant and cosecant?

    sec θ = csc (90° − θ) and csc θ = sec (90° − θ).

  • How do you find a cofunction with the same value as sin 72°?

    Use the identity sin θ = cos (90° − θ), so cofunction is cos 18°.

  • How do you find a cofunction with the same value as csc π/3?

    Use csc θ = sec (90° − θ), so cofunction is sec π/6.

  • What is the angle of elevation?

    The angle between the horizontal line and the line of sight above the observer.

  • What is the angle of depression?

    The angle between the horizontal line and the line of sight below the observer.

  • How do you find the height of a tree given the angle of elevation and shadow length?

    Use tan(angle) = opposite/adjacent where opposite is tree height and adjacent is shadow length.

  • How do you find the altitude increase on a road inclined at an angle after a certain distance?

    Use sin(angle) = opposite/hypotenuse where opposite is altitude increase and hypotenuse is road length.

  • What are the steps to find a reference angle for angles greater than 360° or less than -360°?
    1. Find a positive angle less than 360° coterminal with the given angle.
    2. Draw in standard position.
    3. Use the positive acute angle formed by terminal side and x-axis as reference angle.
  • How do you find the reference angle for θ = 345°?

    Reference angle = 360° − 345° = 15°.

  • How do you find the reference angle for θ = 5π/6?

    Reference angle = π − 5π/6 = π/6.

  • How do you find the reference angle for θ = −135°?

    Reference angle = 135° (absolute value since angle is negative and in standard position).

  • How do you find the reference angle for θ = 580°?

    Find coterminal angle: 580° − 360° = 220°, then reference angle = 220° − 180° = 40°.

  • How do you find the reference angle for θ = 8π/3?

    Find coterminal angle: 8π/3 − 2π = 2π/3, reference angle = π − 2π/3 = π/3.

  • How do you find the reference angle for θ = −13π/6?

    Find coterminal angle: −13π/6 + 4π = 11π/6, reference angle = 2π − 11π/6 = π/6.

  • What is the procedure to use reference angles to evaluate trigonometric functions of any angle θ?
    1. Find the reference angle θ'
    2. Find the function value for θ'
    3. Use the quadrant of θ to assign the correct sign to the function value.
  • How do you evaluate sin 135° using reference angles?

    Reference angle is 45°, sin 135° = sin 45° = \(\frac{\sqrt{2}}{2}\), positive in quadrant II.

  • How do you evaluate cos 4π/3 using reference angles?

    Reference angle is π/3, cos 4π/3 = −cos π/3 = −\(\frac{1}{2}\) (negative in quadrant III).

  • How do you evaluate cot π/3 using reference angles?

    Reference angle is π/3, cot π/3 = \(\frac{1}{\sqrt{3}}\), positive in quadrant I.