Question

The point (5, -2) is on the terminal ray of angle theta, which is in standard position. Without evaluating, explain how you would find the values of six trigonometric functions

Answer 1

To find the six trigonometric functions for the point (5, -2), identify the adjacent and opposite sides, calculate the hypotenuse using the Pythagorean theorem, and use the definitions of trigonometric functions to find their values.

Step-by-step explanation:

To find the values of the six trigonometric functions for the point (5, -2), which is on the terminal ray of angle theta in standard position, we would follow these steps:

1. Identify the x-coordinate (5) as the adjacent side and the y-coordinate (-2) as the opposite side to angle theta.
2. Calculate the hypotenuse using the Pythagorean theorem: hypotenuse = √(x² + y²) = √(5² + (-2)²) = √(29).
3. Determine the six trigonometric functions based on the definitions: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent, csc(theta) = hypotenuse/opposite, sec(theta) = hypotenuse/adjacent, and cot(theta) = adjacent/opposite.
4. Substitute the known values into these ratios to find the trigonometric functions' values.

Note that for all calculations, the angle's direction is not needed as we are using the coordinates to establish the side lengths of the right triangle formed by the point (5, -2) and the origin.

Answer 2

Answer with explanation:

⇒Position of point , which is on the terminal ray of Angle Theta = (5, -2)

⇒Let this point be Represented as P (5, -2) and Origin is Represented by Point O(0,0).

⇒Join OP and draw Perpendicular from P on the X axis,which cut the X axis at A(5,0).

This is Required Right Δ O AP.

⇒Find the Length of OP , using Distance formula.

`D=\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2}\\\\OP=√((5-0)^2+(-2-0)^2)\\\\OP=√(29)`

O A=5 units

PA=2 units

⇒As, the point lies in fourth Quadrant,Only Secant and Cosine function will be positive, all other trigonometric function will have negative value.

By Applying , trigonometric ratio formula,as we have length of three sides of triangle we can evaluate six trigonometric ratios.