Finding Exact Values of Cosine and Tangent for a Given Point in the Coordinate Plane

The problem at hand involves finding the exact values of cosine and tangent of an angle θ, given that the point P(3, -5) lies on the terminal side of this angle in standard position.

Understanding the Problem

The given point is P(3, -5), which lies on the terminal side of the angle θ. This means that if we draw a right triangle with the origin as the vertex, the point P forms the terminal side of the angle θ with the horizontal axis.

Solving for Cosine (cos θ)

To find the exact value of cos θ, we need to use the definition of cosine in a right triangle. In this case, cosine is the ratio of the length of the adjacent side to the hypotenuse.

Steps to Find Cosine

Identify the coordinates of point P: 3 (adjacent side) and -5 (opposite side).

Calculate the length of the hypotenuse using the Pythagorean theorem:

Hypotenuse2 Adjacent2 Opposite2

Hypotenuse2 32 (-5)2

Hypotenuse2 9 25

Hypotenuse2 34

Hypotenuse √34

Use the definition of cosine:

cos θ Adjacent / Hypotenuse

cos θ 3 / √34

Rationalize the denominator:

cos θ 3 / √34 * √34 / √34

cos θ (3√34) / 34

Solving for Tangent (tan θ)

To find the exact value of tan θ, we use the definition of tangent in a right triangle. Tangent is the ratio of the length of the opposite side to the adjacent side.

Steps to Find Tangent

Use the definition of tangent:

tan θ Opposite / Adjacent

tan θ -5 / 3

Understanding the Quadrant

The point P(3, -5) is located in the fourth quadrant (QIV). In QIV, cosine is positive, and tangent is negative.

Nemonic CAST

CAST is a mnemonic for remembering the signs of trigonometric functions in each quadrant:

C osine is positive in QI and QIV, S in is positive in QI and QII, T angent is positive in QI and QIII.

Conclusion

Given the point P(3, -5), we have found the exact values of:
- cos θ (3√34) / 34
- tan θ -5 / 3

It is important to note that while we can find the exact value of tan θ, we cannot provide an exact value for sin θ, as it involves more complex calculations involving arcsin or other methods.