Question

Find the value of each variable.
​

Answer 1

see explanation

Explanation:

Using the exact values of the trigonometric ratios

sin60° =
`(√(3) )/(2)`, cos60° =
`(1)/(2)`

sin45° = cos45° =
`(1)/(√(2) )`

Using the sine ratio on the right triangle on the left

sin60° =
`(opposite)/(hypotenuse)` =
`(a)/(4√(3) )` =
`(√(3) )/(2)`

Cross- multiply

2a = 4
`√(3)` ×
`√(3)` = 12 ( divide both sides by 2 )

a = 6

Using the cosine ratio on the same right triangle

cos60° =
`(adjacent)/(hypotenuse)` =
`(c)/(4√(3) )` =
`(1)/(2)`

Cross- multiply

2c = 4
`√(3)` ( divide both sides by 2 )

c = 2
`√(3)`

------------------------------------------------------------------------------------------

Using the sine/cosine ratios on the right triangle on the right

sin45° =
`(a)/(b)` =
`(6)/(b)` =
`(1)/(√(2) )`

Cross- multiply

b = 6
`√(2)`

cos45° =
`(d)/(b)` =
`(d)/(6√(2) )` =
`(1)/(√(2) )`

Cross- multiply

`√(2)` d = 6
`√(2)` ( divide both sides by
`√(2)` )

d = 6

------------------------------------------------------------------------------------------------

a = 6, b = 6
`√(2)`, c = 2
`√(3)`, d = 6