The terminal side of angle θ lies on the line y=-mx for m>0 and x>0.calculate the values of the trigonometric function
slope of y = -mx is -m, but m is a positive number, so the slope is negative
the slope is also equal to tan Ø
the tangent is negative in quadrants II and IV, but x > 0 in IV
so 270° < Ø < 360°
To calculate the values of the trigonometric functions for an angle θ, whose terminal side lies on the line y = -mx (for m > 0 and x > 0), we will use the properties of right triangles and trigonometric ratios.
Given that the terminal side of angle θ lies on the line y = -mx, we can find the coordinates of the point where this line intersects the x-axis.
Since y = 0 on the x-axis, we substitute y = 0 into the equation of the line to solve for x:
0 = -mx
Divide both sides by -m to isolate x:
0 / -m = x
So, x = 0.
This means that the terminal side of angle θ intersects the x-axis at the point (0, 0).
Now, we can draw a right triangle with one of its acute angles as angle θ, and the leg opposite to angle θ on the y-axis (y-coordinate) and the adjacent leg on the x-axis (x-coordinate).
In this right triangle:
- The y-coordinate is the length of the opposite side (denoted as y).
- The x-coordinate is the length of the adjacent side (denoted as x).
- The hypotenuse is the distance between the origin (0, 0) and the point where the terminal side of angle θ intersects the line y = -mx.
Since the hypotenuse is the length of the line segment connecting the origin (0, 0) and the point on the line y = -mx, we can calculate its length using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, x₁ = 0, y₁ = 0 (the coordinates of the origin), x₂ = x (the x-coordinate of the point on the line), and y₂ = -mx (the y-coordinate of the point on the line).
Plugging in these values, we get:
d = √((x - 0)² + (-mx - 0)²)
= √(x² + m²x²)
= √(1 + m²)x
Now that we have the lengths of the three sides of the right triangle, we can use the trigonometric ratios to find the values of the trigonometric functions.
sin(θ) = opposite/hypotenuse = y/√(1 + m²)x
cos(θ) = adjacent/hypotenuse = x/√(1 + m²)x
tan(θ) = opposite/adjacent = y/x
cosec(θ) = 1/sin(θ) = √(1 + m²)x/y
sec(θ) = 1/cos(θ) = √(1 + m²)x/y
cot(θ) = 1/tan(θ) = x/y