Find the exact value of sin(θ) if θ is in standard position and the terminal side of θ is in Quadrant III and is parallel to the line 7y − 2x + 6 = 0.
slope of the given line = 2/7
then tanθ = 2/7 = y/x ---> x = 7, y = 2 or in III x = -7, y = -2
in your right-angled triangle,
r^2 = 49+4 = 53
r = √53
sinθ = -2/√53
To find the exact value of sin(θ) if the terminal side of θ is parallel to the line 7y − 2x + 6 = 0, we need to determine the angle θ in standard position first.
The equation of the line in slope-intercept form is y = (2/7)x - 6/7.
From the equation, we can see that the slope of the line is (2/7), which corresponds to the tangent of the angle θ.
Since the terminal side of θ is in Quadrant III, the tangent of θ will be negative.
We can find the angle θ using the arctan function. Applying the arctan to both sides of the equation, we get:
arctan(tan(θ)) = arctan(-(2/7))
Since the arctan and tan are inverse functions, they cancel each other out, leaving us with:
θ = arctan(-(2/7))
Now, we need to find the sin(θ).
Since the terminal side of θ is in Quadrant III, the x-coordinate will be negative, and the y-coordinate will be negative.
We know that sin is equal to the y-coordinate divided by the hypotenuse.
To find the hypotenuse, we can use the Pythagorean theorem. From the equation of the line, we have:
(2x - 7y + 6)² + (x² + y²) = 0
Expanding and simplifying, we get:
4x² - 28xy + 49y² + 24x - 84y + 36 + x² + y² = 0
Combining like terms, we get:
5x² - 28xy + 48x + 50y² - 84y + 36 = 0
Since θ is parallel to the line, the ratio of y to x will be the same as the equation of the line, which is (2/7).
Substituting this ratio for y/x, we can simplify the equation:
5x² - 8x² + 48x + 200/49x² - 120/49x + 36 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Where a = 5, b = -8, and c = 36.
Plugging in these values, we can calculate the x-coordinate of the point:
x = (-(-8) ± √((-8)² - 4(5)(36))) / (2(5))
= (8 ± √(64 - 720)) / 10
= (8 ± √(-656)) / 10
Since the square root of a negative number is imaginary, there are no real x-coordinates for this equation.
Therefore, we cannot find the exact value of sin(θ) using this method.