Question

Find the value of each variable. Give exact answers.

Answer 1

See below

Explanation:

Finding a:

`\displaystyle sin (\theta) = (opposite )/(hypotenuse)`

Where θ = 45° , opposite = a, hyp = 4√2

`\displaystyle sin (45)=(a)/(4√(2) ) \\\\(1)/(√(2) ) = (a)/(4√(2) ) \\\\1 = (a)/(4) \\\\\boxed{a = 4}`

Finding c:

`\displaystyle tan (\theta) = (opposite )/(adjacent)`

Where θ = 45°, opposite = 4, adjacent = c

tan 45 = 4 / c

1 = 4 / c

Multiply both sides by c

c = 4

Finding b:

`\displaystyle sin(\theta) = (opposite)/(hypotenuse)`

Where θ = 30°, opposite = a(4), hypotenuse = b

sin 30 = 4 / b

`\displaystyle (1)/(2) = (4)/(b)`

Cross Multiply

1 * b = 4 * 2

b = 8

Finding d:

`\displaystyle tan (\theta) = (opposite )/(adjacent)`

Where θ = 30°, opposite = a(4) , adjacent = d

tan 30 = 4 / d

`\displaystyle (1)/(√(3) ) = (4)/(d) \\\\`

Cross Multiplying

1 * d = 4√3

`\boxed{d = 4√(3)}`

`\rule[225]{225}{2}`

Hope this helped!

~AH1807