If sin θ=5/13 and cos x=4/5
Find cos (θ+x)
So far, I came up with the answer:
(±48 - (±15)) / 65
And I know that the answer was marked out as wrong, did I go wrong with the signs? If so, what signs should I have used, is there a way of telling which quadrant is it in?
Thanks
Unless they say something, I'd assume QI. In that case,
sinθ=5/13 and cosθ=12/13
sinx=3/5 and cosx=4/5
cos (θ+x) = (12/13)(4/5)-(5/13)(3/5) = 33/65
Usually they specify a quadrant if they want to make you determine the sign of the quantities involved.
To find cos(θ+x), we can use the trigonometric identity:
cos(θ+x) = cos(θ)cos(x) - sin(θ)sin(x)
Given that sin θ = 5/13 and cos x = 4/5, we can substitute these values into the formula:
cos(θ+x) = cos(θ)cos(x) - sin(θ)sin(x)
= (5/13)(4/5) - (5/13)(4/5)
Multiplying the fractions, we get:
cos(θ+x) = (20/65) - (20/65)
= 0
So, the correct answer is cos(θ+x) = 0.
Now, let's address the issue of the signs. The signs depend on which quadrant θ and x are located.
Given that sin θ = 5/13, we know that θ is positive. Since sin is positive, this tells us that the angle θ is in either the first or second quadrant.
Given that cos x = 4/5, we know that x is positive. Since cos is positive, this tells us that the angle x is in either the first or fourth quadrant.
To find cos(θ+x), we need to consider the sum of the angles θ and x and determine the quadrant in which the sum falls.
Since both angles are positive, and the answer is 0, this means that the sum of the angles θ and x falls in the fourth quadrant.
Therefore, when evaluating the expression (±48 - (±15)) / 65, the signs should be negative because the angle falls in the fourth quadrant.
Hence, the correct answer is (-48 - (-15)) / 65 = -33/65.