Answer:
The trigonometric ratios are presented below:
[tex]\sin \theta = \frac{AC}{\sqrt{AC^{2} + BC^{2}}}[/tex]
[tex]\cos \theta = \frac{BC}{\sqrt{AC^{2} + BC^{2}}}[/tex]
[tex]\cot \theta = \frac{BC}{AC}[/tex]
[tex]\sec \theta = \frac{\sqrt{AC^{2}+BC^{2}}}{BC}[/tex]
[tex]\csc \theta = \frac{\sqrt{AC^{2}+BC^{2}}}{AC}[/tex]
Step-by-step explanation:
From Trigonometry we know the following definitions for each trigonometric ratio:
Sine
[tex]\sin \theta = \frac{y}{h}[/tex] (1)
Cosine
[tex]\cos \theta = \frac{x}{h}[/tex] (2)
Tangent
[tex]\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}[/tex] (3)
Cotangent
[tex]\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{x}{y}[/tex] (4)
Secant
[tex]\sec \theta = \frac{1}{\cos \theta} = \frac{h}{x}[/tex] (5)
Cosecant
[tex]\csc \theta = \frac{1}{\sin \theta} = \frac{h}{y}[/tex] (6)
Where:
[tex]x[/tex] - Adjacent leg.
[tex]y[/tex] - Opposite leg.
[tex]h[/tex] - Hypotenuse.
The length of the hypotenuse is determined by the Pythagorean Theorem:
[tex]h = \sqrt{x^{2}+y^{2}}[/tex]
If [tex]y = AC[/tex] and [tex]x = BC[/tex], then the trigonometric ratios are presented below:
[tex]\sin \theta = \frac{AC}{\sqrt{AC^{2} + BC^{2}}}[/tex]
[tex]\cos \theta = \frac{BC}{\sqrt{AC^{2} + BC^{2}}}[/tex]
[tex]\cot \theta = \frac{BC}{AC}[/tex]
[tex]\sec \theta = \frac{\sqrt{AC^{2}+BC^{2}}}{BC}[/tex]
[tex]\csc \theta = \frac{\sqrt{AC^{2}+BC^{2}}}{AC}[/tex]
