Suppose sinu = 3/7 and cosu < 0. Use this information to give the exact value of cosu.

Answer 1

Since sin is opposite/hypotenuse, we can establish that the the ratio of opposite to hypotenuse is 3:7.

We can find the length of the adjacent side using the pythagorean theorem:

7² = 3² + x²

x = sqrt(40) = 2*sqrt(10)

since cos is adjacent/hypotenuse, we can say that cos = 2*sqrt(10)/7.

Since we were given that cos(u) < 0, we negate our answer, so

cos(u) = -2*sqrt(10)/7

Sign just changes which quadrant you draw your triangle in. From sin(u) = 3/7, we know the y value of the triangle is positive, so we know we are in either quadrant I or II. We know that since cos(u) < 0, we are in quadrant II, where the x value of the triangle is negative. If cos(u) were greater than zero, cos(u) would be 2*sqrt(10)/7 and we would be in quadrant I, where the x value of the triangle is positive.

Answer 2

When considering the coordinate axis, sin is positive when y is positive, and cos is positive when x is positive. As cos(u) is negative, we know that the x value is negative.

Furthermore, for a right triangle we know that a² + b² = c²

Here, a = 3, c = 7, and b is unknown.

b = √(c²-a²) = √(7²-3²) = √(49 - 9 ) = √40 = 6.325

a is the height of the triangle, b is the width. As sin is height over hypotenuse (a/c) and cos is width over hypotenuse (b/c), cos = (6.325/7)

In our case as we know x is negative, cos = (-6.325/7) or -0.9036

Hope that helps!