Did you know?
In trigonometry, the primary trigonometric ratios are essential for understanding the relationships between angles and sides in a right triangle. Given that angle θ, in standard position, has a terminal arm with the point 𝑃(5, −3), we can determine the exact values for all three primary trig ratios:
1. The sine of θ (sin θ) is equal to the y-coordinate of the terminal arm's point divided by the length of the radius. In this case, sin θ = -3/√(5^2 + (-3)^2) = -3/√34.
2. The cosine of θ (cos θ) is equal to the x-coordinate of the terminal arm's point divided by the length of the radius. In this case, cos θ = 5/√(5^2 + (-3)^2) = 5/√34.
3. The tangent of θ (tan θ) is equal to the sine of θ divided by the cosine of θ. Thus, tan θ = (-3/√34) / (5/√34) = -3/5.
Furthermore, by using the coordinates of point 𝑃(5, −3), we can find the angle θ using inverse trigonometric functions. The angle θ, measured counterclockwise from the positive x-axis, can be found using the equation θ = arctan(-3/5) ≈ -30.96 degrees. However, since the angle must be between 0 and 360 degrees, we can add 360 degrees to obtain the positive equivalent, giving us θ ≈ 329.04 degrees.
