Question

Write each ratio in simplest form.

sin X=
cos X=
sin Y=
cos Y =
tan X =
tan Y =
X
27
N
45
36
Y

Answer 1

`\sf sin X = (4)/(5)`

`\sf cos X =(3)/(5)`

`\sf sin Y = (3)/(5)`

`\sf cos Y =(4)/(5)`

`\sf tan X =(4)/(3)`

`\sf tan Y =(3)/(4)`

Explanation:

Given:

In ∆ XYZ

• XZ = 27
• YZ = 36
• XY = 45

To find:

• sin X = ?
• cos X = ?
• sin Y = ?
• cos Y = ?
• tan X = ?
• tan Y = ?

Solution:

To find the sine, cosine, and tangent of angles in a triangle, we can use the following formulas:

sin X = opposite over hypotenuse

cos X = adjacent over hypotenuse

tan X = opposite over adjacent

sin X

The opposite side to angle X is YZ, and the hypotenuse is XY.

Therefore, the sine of angle X is:

`\sf sin X = (opposite)/(hypotenuse)\\\\ =(YZ )/( XY) \\\\ = (36 )/( 45) \\\\ = (4)/(5)`

cos X

The adjacent side to angle X is XZ, and the hypotenuse is XY.

Therefore, the cosine of angle X is:

`\sf cos X =(adjacent)/(hypotenuse)\\\\ = ( XZ )/( XY) = (27 )/( 45) \\\\ = ( 3)/(5)`

sin Y

The opposite side to angle Y is XZ, and the hypotenuse is XY.

Therefore, the sine of angle Y is:

`\sf sin Y = (opposite)/(hypotenuse)\\\\ = ( XZ )/( XY) = (27 )/( 45) \\\\ = ( 3)/(5)`

cos Y

The adjacent side to angle Y is YZ, and the hypotenuse is XY.

Therefore, the cosine of angle Y is:

`\sf cos Y = (adjacent)/(hypotenuse)\\\\ =(YZ )/( XY) \\\\ = (36 )/( 45) \\\\ = (4)/(5)`

tan X

The opposite side to angle X is YZ, and the adjacent side is XZ.

Therefore, the tangent of angle X is:

`\sf tan X = (opposite)/(adjacent)\\\\ = (YZ )/(XZ)\\\\ = (36 )/( 27) \\\\ (4)/(3)`

tan Y

The opposite side to angle Y is XZ, and the adjacent side is YZ.

Therefore, the tangent of angle Y is:

`\sf tan Y = (opposite)/(adjacent)\\\\ = (XZ )/(YZ)\\\\ = (27 )/( 36) \\\\ (3)/(4)`

Answers:

`\sf sin X = (4)/(5)`

`\sf cos X =(3)/(5)`

`\sf sin Y = (3)/(5)`

`\sf cos Y =(4)/(5)`

`\sf tan X =(4)/(3)`

`\sf tan Y =(3)/(4)`