If tan theta=-3/4 and 90°<theta<180° Then find cos theta

So you are told that Ø is in quadrant II, and it looks like you are dealing with the 3-4-5 right-angled triangle

so y = 3, x = -4, r = 5
cosØ = -4/5

To find cos theta, we can use the given information about tan theta.

First, let's understand the relationship between tan theta and cos theta. The tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle. In terms of the sides, tan theta = opposite/adjacent.

To find cos theta, we need to figure out the adjacent side length and the hypotenuse length of the right triangle.

In this case, since we know that tan theta = -3/4 and 90° < theta < 180°, we can assume that theta is in the second quadrant, where the tangent is negative.

Let's consider a right triangle in the second quadrant:

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Since tan theta is negative, that means the opposite side length must be negative. Therefore, let's assign the opposite side as -3 and the adjacent side as 4. This will satisfy the condition tan theta = -3/4.

Using the Pythagorean theorem, we can find the hypotenuse length. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the two legs. So, in this case, we have:

hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = (-3)^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25

Now, we take the square root of both sides to find the hypotenuse length:

hypotenuse = √25
hypotenuse = 5

Now, we have the lengths of the adjacent side and the hypotenuse.

Next, we can find cos theta. The cosine function is defined as the ratio of the adjacent side to the hypotenuse.

cos theta = adjacent/hypotenuse
cos theta = 4/5

Therefore, cos theta is equal to 4/5.