Find the value of the sin function if tan = 5/4 and cos < 0

Answer 1

We can use the Pythagorean identity to find the value of sin when tan = 5/4.

Given:
tan = 5/4
cos < 0

Using the formula for tan:
tan = sin/cos

sin/cos = 5/4

We also know that cos < 0. Since cos = adjacent/hypotenuse, this means that the adjacent side is negative and the hypotenuse is positive.

Let's assume that the opposite side is represented by "y" and the adjacent side is represented by "-x" (negative because cos is negative). Then, using the Pythagorean theorem:

(x^2 + y^2) = (4^2)

(x^2 + y^2) = 16

Since tan = sin/cos, we know that sin = tan * cos:

sin = (5/4) * (-x)

sin = -5x/4

Now, we can use the Pythagorean identity to solve for sin using the values we have:

(sin^2 + cos^2) = 1

((-5x/4)^2 + (-x)^2) = 1

(25x^2/16 + x^2) = 1

(25x^2 + 16x^2) = 16

41x^2 = 16

x^2 = 16/41

x = ±√(16/41)

Since cos is negative, we take the negative value, so:

x = -√(16/41)

Now we can substitute this value back into sin = -5x/4:

sin = -5(-√(16/41))/4

sin = 5√(16/41)/4

Finally, we simplify:

sin ≈ 0.622

Answer 2

What a lot of work!
For our reference angle θ, if tanθ = 5/4, then
y = 5
x = 4
so r = √41
Now, if tanθ > 0 and cosθ < 0, we are in QIII, so
y = -5
x = -4
r = √41
That means sinθ = y/r = -5/√41

Answer 3

Apologies for the extra steps in the previous response. You are correct.

Given that tanθ = 5/4 and cosθ < 0, we are in Quadrant III. Therefore, y = -5, x = -4, and r = √41.

Using the definition of sinθ as y/r, we can calculate sinθ:

sinθ = y/r = -5/√41 ≈ -0.786