Question

In the standard \((x, y)\) coordinate plane below, an angle is shown whose vertex is the origin. One side of this angle with measure \(\theta\) passes through \((4,-3),\) and the other side includes the positive \(x\) -axis. What is the cosine of \(\theta ?\) F. \(-\frac{4}{3}\) G. \(-\frac{3}{4}\) H. \(-\frac{3}{5}\) J. \(\frac{4}{5}\) K. \(\frac{5}{4}\)

Short answer

Answer: \(\frac{3}{5}\).

Step-by-step solution

Find the distance between point (4, -3) and the origin

In order to find the length of the hypotenuse of the right triangle formed by the angle \(\theta\), we will need to find the distance between the point \((4, -3)\) and the origin \((0, 0)\). We can use the distance formula for this: $$D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{(4-0)^2 + (-3-0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.$$ So the distance between the point \((4, -3)\) and the origin is 5.

Determine the length of the sides adjacent and opposite to the angle

Since the right triangle formed has one side that is vertical (from \((4, -3)\) to \((4, 0)\)) and one side that is horizontal (from \((4, 0)\) to the origin \((0, 0)\)), the length of the side opposite to the angle \(\theta\) is 4 and the length of the side adjacent to the angle is 3 (the distance from \((0, 0)\) to \((4, 0)\)).

Calculate the cosine of the angle

We can now find the cosine of the angle \(\theta\). Recall that cosine is defined as the length of the adjacent side divided by the length of the hypotenuse: $$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{5}.$$ Thus, the correct answer is H. \(-\frac{3}{5}\).

Key concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. One of the key functions in trigonometry is the cosine, which is a ratio that compares the length of the adjacent side of an angle to the hypotenuse in a right triangle. In the context of a coordinate plane, the cosine of an angle can help determine a point's location in relation to the origin based on the angle it makes with the positive x-axis.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface divided into four quadrants by a vertical line called the y-axis and a horizontal line called the x-axis. The point where these axes intersect is called the origin. In trigonometry, the coordinate plane is used to visualize angles and their relationships to points. For example, if an angle's side passes through a point like (4, -3), you can use this information—along with the origin (0,0)—to create a right triangle and solve trigonometric problems such as finding the cosine of the angle.

Distance Formula

The distance formula is a mathematical equation used to calculate the straight-line distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is expressed as:
\(D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
This formula is essential for finding the length of the hypotenuse in a right triangle when the coordinates of its vertices are known, as it is the case with the point (4, -3) and the origin (0, 0). The distance calculated serves as the hypotenuse when determining trigonometric ratios.

Right Triangle

A right triangle is a triangle that has one 90-degree angle, which is essential for many trigonometric calculations. In relation to a coordinate plane, one can often use the axes as references to create right triangles where the hypotenuse extends from the origin to a given point and the legs are parallel to the axes. This scenario simplifies the process of calculating trigonometric functions such as cosine, sine, and tangent, using the lengths of the sides of these triangles.