cos theta = 5/13 with -pi/2 < theta <0 find each of the following
a. sin theta (- 5pi/4)
help plz?
To find each of the following expressions, we will use the given value of θ = -5π/4 and the given trigonometric equation cos(θ) = 5/13. We'll start with finding sin(θ).
a) sin(θ):
To find sin(θ), we can use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
Given that cos(θ) = 5/13, we can plug it into the equation: sin^2(θ) + (5/13)^2 = 1.
Rearranging the equation, we have: sin^2(θ) = 1 - (25/169) = 144/169.
Taking the square root of both sides, we get: sin(θ) = ±√(144/169) = ±(12/13).
Now, substituting θ = -5π/4, we can determine the sign of sin(θ) based on the given interval for θ. Since -π/2 < θ < 0, we are in the third quadrant where sin(θ) is negative.
Therefore, sin(θ) = -12/13.
So, a) sin(θ) (- 5π/4) = -12/13.