Question

Find the value of each variable provide proofs​

Answer 1

both sides are equal to:
`8√(2)`

Explanation:

Trigonometric Functions:

Trigonometric functions are defined as the ratios between sides based on some angle. There's three main trig functions you want to remember which are defined as:

`sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\\\\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\\\\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}`

and then there are three other trig functions, which are simply the reciprocals:

`csc(\theta)= \frac{\text{hypotenuse}}{\text{opposite}}=(1)/(sin(\theta))\\\\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}=(1)/(cos(\theta))\\\\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}=(1)/(tan(\theta))`

Now in our case we have the angle of 45 degrees and we have the hypotenuse with length 16, and we want to either solve for the opposite (x) or adjacent side (y)

Let's solve for the adjacent side, in which case we want a trig function defined using the hypotenuse and adjacent side, which is our cosine function.

`cos(45)=(y)/(16)\implies cos(45)*16=y` and from here we can use a calculator or the unit circle which gives us an exact value. Using the unit circle, we can determine that:
`cos(45)=(√(2))/(2)` and plugging that into our equation we get:
`y=(√(2))/(2)*16=8√(2)` and we can use a calculator to approximate this: 11.313

Now we can also use the trig functions to find "x", which is the opposite side. In this case we want to use a trig function which is defined using the opposite and hypotenuse side, which is our sine function:
`sin(45)=(x)/(16)\implies sin(45)*16=x` and the cool thing about this, is that sin(45) and cos(45) are actually equal, so our "x" and "y' side are exactly equal:
`x=8√(2)`

We can verify this using the Pythagorean Theorem:
`(8√(2))^2+(8√(2))^2=16^2\\\\(8^2*√(2)^2)+(8^2+√(2)^2)=256\\\\(64*2)+(64*2)=256\\\\128+128=256\\\\256=256`