Final answer:
To determine the equation of the line that makes a specific angle with the x-axis, the sine of the angle is used to calculate the slope using trigonometric identities. Then, the point-slope form of the line's equation is applied using the slope and the given point, leading to the linear equation y = (3/4)x + 5/4.
Step-by-step explanation:
To find the equation of the straight line which passes through the point (1, 2) and makes an angle with the positive direction of the x-axis such that the sine of this angle is 3/5, we first need to find the slope of the line.
The slope (m) of a line is the tangent of the angle it makes with the positive x-axis.
Since we are given the sine of the angle, we can use the trigonometric identity sin2θ + cos2θ = 1 to find the cosine, and subsequently, the tangent (since tan θ = sin θ / cos θ).
In this case, sin θ = 3/5, so we can calculate cos θ = √(1 - (3/5)2) = 4/5.
Thus, the slope (m) is m = sin θ / cos θ = (3/5) / (4/5) = 3/4.
Now, using the point-slope form of the line's equation, y - y1 = m(x - x1), where (x1, y1) is a known point on the line and m is the slope, we substitute the known point (1, 2) and the slope m = 3/4 to get:
y - 2 = (3/4)(x - 1)
Finally, simplifying the equation, we get:
y = (3/4)x + 5/4