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Precalculus: Analytic Trigonometry - Double-Angle and Half-Angle Formulas

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이 집합의 용어 (20)

  • What is the double-angle formula for cosine?

    cos(2θ) = cos²θ − sin²θ

  • State the double-angle formula for sine.

    sin(2θ) = 2 sinθ cosθ

  • How can you express cos(2θ) using only cosine?

    cos(2θ) = 2 cos²θ − 1

  • How can you express cos(2θ) using only sine?

    cos(2θ) = 1 − 2 sin²θ

  • What is the half-angle formula for sine?

    sin(θ/2) = ±√((1 − cosθ)/2), sign depends on quadrant of θ/2

  • What is the half-angle formula for cosine?

    cos(θ/2) = ±√((1 + cosθ)/2), sign depends on quadrant of θ/2

  • How do you find the exact value of sin(2θ) if sinθ and cosθ are known?

    Use sin(2θ) = 2 sinθ cosθ and substitute the known values.

  • How is the range R of a projectile related to the angle θ and initial velocity v?

    R = (v² sin(2θ)) / g, where g is acceleration due to gravity.

  • At what angle θ is the range R of a projectile maximized?

    The range is maximized at θ = 45°.

  • How do you solve trigonometric equations involving double-angle formulas?

    Rewrite the equation using double-angle identities, then solve for the angle considering the general solution.

  • What is the half-angle formula for tangent?

    tan(θ/2) = ±√((1 − cosθ)/(1 + cosθ)), or tan(θ/2) = sinθ / (1 + cosθ), sign depends on quadrant.

  • How do you determine the sign in half-angle formulas?

    The sign (+ or −) depends on the quadrant in which the half-angle lies.

  • What is the identity to express sin²θ in terms of cos(2θ)?

    sin²θ = (1 − cos(2θ)) / 2

  • What is the identity to express cos²θ in terms of cos(2θ)?

    cos²θ = (1 + cos(2θ)) / 2

  • How can you rewrite sin²θ or cos²θ to avoid powers greater than 1?

    Use power-reducing formulas derived from double-angle identities.

  • What is the general solution for an equation involving sin(2θ) = k?

    θ = (1/2) arcsin(k) + nπ, where n is any integer.

  • How do you find exact values of trigonometric functions using half-angle formulas?

    Apply the half-angle formulas with known cosine or sine values and determine the correct sign based on the quadrant.

  • What is the relationship between the coordinates (x, y) on a circle and the angle θ?

    For a circle of radius r, x = r cosθ and y = r sinθ.

  • How do you find the length of the side adjacent to an angle θ in a right triangle on a circle?

    Use x = r cosθ, where r is the radius.

  • What is the formula for the distance from the origin to a point (x, y) on a circle?

    r = √(x² + y²)