Answer:
[tex]\sqrt{3}:3\\1:\sqrt{3}[/tex]
Step-by-step explanation:
Trigonometric Ratios
The ratios of the sides of a right triangle are called trigonometric ratios. There are six trigonometric ratios, sine, cosine, tangent, cosecant, secant, and cotangent.
The longest side of the triangle is called the hypotenuse and the other two sides are the legs.
Selecting any of the acute angles as a reference, it has an adjacent side and an opposite side. The trigonometric ratios are defined upon those sides.
The tangent ratio is defined as:
[tex]\displaystyle \tan\theta=\frac{\text{opposite leg}}{\text{adjacent leg}}[/tex]
The cotangent ratio is defined as:
[tex]\displaystyle \cot\theta=\frac{\text{adjacent leg}}{\text{opposite leg}}[/tex]
As shown above, the tangent and the cotangent are the ratios of the two legs in any possible order.
If the triangle has angles of 30°, 60°, and 90°, then we can take one of the acute angles and find the tangent and cotangent to know the required ratios:
Let's take for example the 30° angle:
[tex]\displaystyle \text{ratio of legs: }\tan 30^\circ=\frac{\sqrt{3}}{3}[/tex]
[tex]\displaystyle \text{ratio of legs: }\cot 30^\circ=\sqrt{3}[/tex]
Thus the possible ratios from the list are:
[tex]\sqrt{3}:3\\1:\sqrt{3}[/tex]
The reciprocals of those ratios are also applicable
