The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

Substitute the name of each side into the definition of the cosine function.

Set up the equation to solve for the adjacent side, in this case .

Substitute the values of each variable into the formula for cosine.

Multiply .

Combine and .

Combine and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Simplify the expression.

Multiply by .

Divide by .

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

Solve the equation for .

Substitute the actual values into the equation.

Simplify the expression.

Apply the product rule to .

Raise to the power of .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Simplify the expression.

Multiply by .

Raise to the power of .

Multiply by .

Subtract from .

Rewrite as .

Pull terms out from under the radical.

The absolute value is the distance between a number and zero. The distance between and is .

These are the results for all angles and sides for the given triangle.

Solve the Triangle tri()(45)(9 square root of 2)(45)()(90)