The six trigonometric functions related to (x, y) = (20, 48): sin θ = 12/13, cos θ = 5/13, tan θ = 12/5, cot θ = 5/12, sec θ = 13/5, csc θ = 13/12.
How to determine the six trigonometric functions from coordinates in rectangular form
Given the coordinates in rectangular form, that is, (x, y), where each component belongs to an orthogonal axis of reference. By trigonometry we have six functions to be calculated:
[tex]\sin \theta = \frac{y}{\sqrt{x^{2}+y^{2}}}[/tex] (1)
[tex]\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}[/tex] (2)
[tex]\tan \theta = \frac{\sin \theta}{\cos \theta}[/tex] (3)
[tex]\cot \theta = \frac{1}{\tan \theta}[/tex] (4)
[tex]\sec \theta = \frac{1}{\cos \theta}[/tex] (5)
[tex]\csc \theta = \frac{1}{\sin \theta}[/tex] (6)
If we know that x = 20 and y = 48, then the six trigonometric functions are, respectively:
[tex]\sin \theta = \frac{48}{\sqrt{20^{2}+48^{2}}}[/tex]
[tex]\sin \theta = \frac{12}{13}[/tex]
[tex]\cos \theta = \frac{20}{\sqrt{20^{2}+48^{2}}}[/tex]
[tex]\cos \theta = \frac{5}{13}[/tex]
[tex]\tan \theta = \frac{12}{5}[/tex]
[tex]\cot \theta = \frac{5}{12}[/tex]
[tex]\sec \theta = \frac{13}{5}[/tex]
[tex]\csc \theta = \frac{13}{12}[/tex]
The six trigonometric functions related to (x, y) = (20, 48): sin θ = 12/13, cos θ = 5/13, tan θ = 12/5, cot θ = 5/12, sec θ = 13/5, csc θ = 13/12. [tex]\blacksquare[/tex]
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