Question

use the diagram of triangle XYZ to answer the questions. what is the length of side XY? what is the value of sin (X)? what is the value of cos (X)? what is the value of tan (X)?

Answer 1

The missing diagram is attached below

Answer:

The length of XY is 10 units

The value of sin (X) is
`(4)/(5)`

The value of cos (X) is
`(3)/(5)`

The value of tan (X) is
`(4)/(3)`

Explanation:

In Δ XYZ

∵ ∠Z is a right angle

∴ XY is the hypotenuse

∴ XZ and YZ are the legs of the triangle

- Use Pythagoras Theorem to find the length of XY

∵ (XY)² = (XZ)² + (YZ)²

∵ XZ = 6 units

∵ YZ = 8 units

- Substitute them in the formula above

∴ (XY)² = (6)² + (8)²

∴ (XY)² = 36 + 64

∴ (XY)² = 100

- Take √ for both sides

∴ XY = 10 units

In The right Δ there are trigonometry ratios

sin X =
`(opposite)/(hypotenuse)`

cos X =
`(adjacent)/(hypotenuse)`

tan X =
`(opposite)/(adjacent)`

∵ The side opposite to ∠X is YZ

∵ The side adjacent to ∠ X is XZ

∵ The hypotenuse is XY

∴ sin X =
`(YZ)/(XY)`

∴ cos X =
`(XZ)/(XY)`

∴ tan X =
`(YZ)/(XZ)`

∵ YZ = 8 and XY = 10

∴ sin X =
`(8)/(10)`

- Reduce it to its lowest term

∴ sin (X) =
`(4)/(5)`

∴ cos X =
`(6)/(10)`

- Reduce it to its lowest term

∴ cos (X) =
`(3)/(5)`

∴ tan X =
`(8)/(6)`

- Reduce it to its lowest term

∴ tan (X) =
`(4)/(3)`