Solve tan(40/90) for the double angle cos(2theta). I know I use the x^2 + y^2 = r^2 formula and cos(2theta) = cos^2theta - sin^2theta. But I don't understand my answer
tan(40/90) ?
That's an odd notation. Is there some reason why you did not say 4/9? Evan that seems unusual for such a problem.
To solve tan(40/90), we need to first simplify the expression by finding the tangent of the angle. The tangent of an angle is calculated by dividing the length of the opposite side by the length of the adjacent side in a right triangle.
For tan(40/90), let's assume we have a right triangle with angle A, where the length of the opposite side is 40 and the adjacent side is 90. To find the length of the hypotenuse side, we can use the Pythagorean theorem: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
In this case, the equation becomes 40² + 90² = c².
Solving this equation, we find that c² = 3600 + 8100 = 11700. Taking the square root of both sides, we get c ≈ √11700 ≈ 108.11.
Now that we have the values for all three sides of the right triangle (opposite, adjacent, and hypotenuse), we can find the tangent of angle A using the equation tan(A) = opposite/adjacent.
tan(A) = 40/90 ≈ 0.4444.
Now to find cos(2theta), we'll use the double angle formula: cos(2theta) = cos²(theta) - sin²(theta).
Since we know the value of tan(A) = 0.4444, we can find the values of sin(A) and cos(A) using the definitions: sin(A) = opposite/hypotenuse and cos(A) = adjacent/hypotenuse.
sin(A) = 40/108.11 ≈ 0.3691
cos(A) = 90/108.11 ≈ 0.8315
Now, substitute the values of sin(A) and cos(A) into the double angle formula:
cos(2A) = cos²(A) - sin²(A)
= (0.8315)² - (0.3691)²
= 0.6923 - 0.1363
≈ 0.556.
Therefore, cos(2theta) ≈ 0.556.