Question

I need to know how to solve trigonometric ratios. Need to determine the length for the unknown sides

Answer 1

Given:

AB=9cm, AC=y, BC=z, and the angle ABC = 61.2 degrees.

XZ=21.3ft, XY=x, YZ=y, and the angle XYZ = 66.4 degrees.

Required:

We need to find unknown sides.

Explanation:

Consider the triangle ABC.

AB is the hypotenuse, BC is the adjacent side, and AC is the opposite side.

Use sine formula.

$sin\theta=\frac{opposite\text{ side}}{hypotenuse}$$Substitute\text{ }\theta=61.2\degree\text{ opposite side =y, and hypotenuse = 9cm in the formula.}$$sin61.2\degree=\frac{y}{9}$$y=9sin61.2\degree$$y=7.9\text{ cm}$

Use the cosine formula.

$cos\theta=\frac{adjacent\text{ side}}{hypotenuse}$$Substitute\text{ }\theta=61.2\degree\text{ adjacent side =x, and hypotenuse = 9cm in the formula.}$$cos61.2\degree=\frac{x}{9}$$x=9cos61.2\degree$$x=4.3cm$

We get x =4.3 cm and y = 7.9 cm.

Consider the triangle XYZ.

YZ is the hypotenuse, XY is the adjacent side, and XZ is the opposite side.

Use sine formula.

$sin\theta=\frac{opposite\text{ side}}{hypotenuse}$$Substitute\text{ }\theta=66.4\degree\text{ opposite side =21.3, and hypotenuse = y in the formula.}$$sin66.4=\frac{21.3}{y}$$y=\frac{21.3}{sin66.4}$$y=23.2ft$

Use the cosine formula.

$cos\theta=\frac{adjacent\text{ side}}{hypotenuse}$$Substitute\text{ }\theta=66.4\degree\text{ adjacent side =x, and hypotenuse = 23.2 in the formula.}$$cos66.4\degree=\frac{x}{23.2}$$x=23.2cos66.4\degree$$x=9.3ft$

Final answer:

1)

$x=4.3cm$

$y=7.9\text{ cm}$

2)

$x=9.3ft$

$y=23.2ft$