Given that π < 𝑥 < 2π and tan 𝑥 = 3, determine the exact value of cos(2𝑥).
in QIII, draw a standard triangle with legs 3 and 1. It should be clear that
cosθ = x/r = -3/√10
cos(2x) = 2cos^2(x) - 1 = 2 * 9/10 - 1 = 4/5
should be -4/5
should have been:
cosθ = x/r = -1/√10
cos(2x) = 2cos^2(x) - 1 = 2 * 1/10 - 1 = - 4/5
(caught your typo by doing cos (2x) = cos^2 x - sin^2 x)
To find the exact value of cos(2𝑥), we need to make use of trigonometric identities. One such identity is the double-angle formula for cosine, which states:
cos(2𝑥) = cos²(𝑥) - sin²(𝑥)
First, we need to find the values of cos(𝑥) and sin(𝑥). We already know that tan(𝑥) = 3, and tangent is defined as the ratio of sin(𝑥) to cos(𝑥):
tan(𝑥) = sin(𝑥) / cos(𝑥)
Since 𝑡𝑎𝑛𝜃 is positive in the given range (0 < 𝜃 < 𝜋), we conclude that both sin(𝑥) and cos(𝑥) are positive. Now we can solve for sin(𝑥) and cos(𝑥) by using the Pythagorean identity:
sin²(𝑥) + cos²(𝑥) = 1
Plugging in the given information, we have:
3² + cos²(𝑥) = 1
9 + cos²(𝑥) = 1
cos²(𝑥) = 1 - 9
cos²(𝑥) = -8
This is not possible since cosine cannot be negative. Therefore, there is no solution for cos(𝑥) in the given range. As a result, we cannot determine the exact value of cos(2𝑥) in this case.