Given that π < 𝑥 < 2π and tan 𝑥 = 3/4, determine the exact value of cos(2𝑥).
First APPROXIMATE
𝑥 must be in quadrant three, lower left, because cos and sin have the same sign to get the positive tangent.
so 𝑥 is about 180 + tan^-1 3/4 = 180 + 36.86 deg = about 216.86 deg
so 2 𝑥 = about 433.7 deg
433.7 - 360 = 73.7 degrees, in quadrant one so approximately 0.28
now EXACT: (note 3,4,5 triangle)
cos 2𝑥 = cos^2 𝑥 - sin^2 x = (-4/5)^2 - (-3/5)^2
= 16/25 - 9/25 = 7/25 YES, about 0.28
Cos 2x = 4/5 and 90° < x < 180° find six trignometric function?
To determine the exact value of cos(2𝑥), we'll use the double-angle identity for cosine, which states that:
cos(2𝑥) = cos²(𝑥) - sin²(𝑥)
To find cos(𝑥) and sin(𝑥), we need to find the ratio of the legs of a right triangle. The given information tells us that tan(𝑥) = 3/4.
We can label one leg as 3 (opposite side) and the other leg as 4 (adjacent side). Using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse² = (opposite side)² + (adjacent side)²
hypotenuse² = 3² + 4²
hypotenuse² = 9 + 16
hypotenuse² = 25
hypotenuse = 5
Therefore, in the right triangle, the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.
Now we can evaluate cos(𝑥) and sin(𝑥):
cos(𝑥) = (adjacent side) / (hypotenuse) = 4/5
sin(𝑥) = (opposite side) / (hypotenuse) = 3/5
Now that we have cos(𝑥) and sin(𝑥) values, we can substitute them to find cos(2𝑥):
cos(2𝑥) = cos²(𝑥) - sin²(𝑥)
cos(2𝑥) = (4/5)² - (3/5)²
cos(2𝑥) = 16/25 - 9/25
cos(2𝑥) = 7/25
Therefore, the exact value of cos(2𝑥) is 7/25.