1) cos (θ / 2) = √[(1 + cos θ) / 2], sin (θ / 2) = √[(1 - cos θ) / 2], tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) (x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°). The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) The <em>linear</em> function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
<h3>How to apply trigonometry on deriving formulas and transforming points</h3>
1) The following <em>trigonometric</em> formulae are used to derive the <em>half-angle</em> formulas:
sin² θ / 2 + cos² θ / 2 = 1 (1)
cos θ = cos² (θ / 2) - sin² (θ / 2) (2)
First, we derive the formula for the sine of a <em>half</em> angle:
cos θ = 2 · cos² (θ / 2) - 1
cos² (θ / 2) = (1 + cos θ) / 2
cos (θ / 2) = √[(1 + cos θ) / 2]
Second, we derive the formula for the cosine of a <em>half</em> angle:
cos θ = 1 - 2 · sin² (θ / 2)
2 · sin² (θ / 2) = 1 - cos θ
sin² (θ / 2) = (1 - cos θ) / 2
sin (θ / 2) = √[(1 - cos θ) / 2]
Third, we derive the formula for the tangent of a <em>half</em> angle:
tan (θ / 2) = sin (θ / 2) / cos (θ / 2)
tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) The formulae for the conversion of coordinates in <em>rectangular</em> form to <em>polar</em> form are obtained by <em>trigonometric</em> functions:
(x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) Let be the point (x, y) = (2, 3), the coordinates in <em>polar</em> form are:
r = √(2² + 3²)
r = √13
θ = atan(3 / 2)
θ ≈ 56.309°
The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°).
Let be the point (r, θ) = (4, 30°), the coordinates in <em>rectangular</em> form are:
(x, y) = (4 · cos 30°, 4 · sin 30°)
(x, y) = (2√3, 2)
The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) Let be the <em>linear</em> function y = 5 · x - 8, we proceed to use the following <em>substitution</em> formulas: x = r · cos θ, y = r · sin θ
r · sin θ = 5 · r · cos θ - 8
r · sin θ - 5 · r · cos θ = - 8
r · (sin θ - 5 · cos θ) = - 8
r = - 8 / (sin θ - 5 · cos θ)
The <em>linear</em> function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
To learn more on trigonometric expressions: brainly.com/question/14746686
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