Given the right triangle shown below, prove cos2(θ) + sin2(θ) = 1. A triangle with sides x, y, and hypotenuse r. Angle theta is opposite to side y.

By
, cos(θ) = StartFraction x Over r EndFraction, and by
, sin(θ) = StartFraction y Over r EndFraction.

Multiplying both sides of the above equations by r, we get that x = r cos(θ) and y = r sin(θ).

The
states that x2 + y2 = r2.

By
, we have [r cos(θ)]2 + [r sin(θ)]2 = r2.

Applying the
, the equation can be written as r2 cos2(θ) + r2 sin2(θ) = r2.

Dividing both sides of the equation by r2 results in cos2(θ) + sin2(θ) = 1.

I've been working on this for a few days and i just don't understand, it's due in a few hours. you. the direction of a vector is defined as the angle of the vector in relation to a horizontal line. as a standard, this angle is measured counterclockwise from the positive x-axis. the direction or angle of v in the diagram is α.part a: how can you use trigonometric ratios to calculate the direction α of a general vector v = < x, y> similar to the diagram? part bsuppose that vector v lies in quadrant ii, quadrant iii, or quadrant iv. how can you use trigonometric ratios to calculate the direction (i. e., angle) of the vector in each of these quadrants with respect to the positive x-axis? the angle between the vector and the positive x-axis will be greater than 90 degrees in each case. part cnow try a numerical problem. what is the direction of the vector w = < -1, 6 > ?