Trigonometric Functions #6 - Questions and Answers

Question 1 Determine the surface for which the equation is given. ρ = sinθsinφ Answer To see what's going on here, transform the equation in the following way. ρ = sinθsinφ ρ² = ρsinθsinφ Next, recall that ρ² = x² + y² + z² and that y = ρsinθsinφ. Our equation is actually: x² + y² + z² = y Complete the square involving y to get: x² + (y - 1/2)² + z² = 1/4 This is a sphere of radius 1/2 centered at (0, 1/2, 0). Result: This is a sphere of radius 1/2 centered at (0, 1/2, 0). Answer 2 The given equation is ρ = sin(θ)sin(φ). This equation represents a mathematical surface in spherical coordinates. The variables θ and φ represent the spherical angles, and ρ represents the radial distance from the origin to a point on the surface. In spherical coordinates, θ represents the azimuthal angle (measured from the positive x-axis in the xy-plane), and φ represents the polar angle (measured from the positive z-axis). To visualize this surface, we can consider different values of θ and φ and calculate the corresponding values of ρ using the equation. For example, if we fix θ = π/2 and vary φ from 0 to 2π, we can plot the resulting values of ρ. Similarly, we can fix φ = π/2 and vary θ to obtain another plot. The resulting surface will depend on the range of θ and φ values chosen. The equation ρ = sin(θ)sin(φ) describes a surface that exhibits symmetry with respect to the xz-plane and the yz-plane. It is important to note that the range of θ and φ values should be appropriately chosen to obtain a meaningful representation of the surface. Answer 3 Explanation: To determine the surface for the given equation ρ = sin(θ)sin(φ), we can use spherical coordinates. In spherical coordinates, ρ represents the radial distance from the origin, θ represents the polar angle measured from the positive z-axis, and φ represents the azimuthal angle measured from the positive x-axis. To find the surface, we can express the equation in terms of spherical coordinates. Since ρ = sin(θ)sin(φ), we can substitute the spherical coordinate expressions into the equation: ρ = sin(θ)sin(φ) r = ρsin(θ) = sin(θ)sin(φ)sin(θ) = sin²(θ)sin(φ) Using the relationship between Cartesian coordinates (x,y,z) and spherical coordinates (r, θ, φ), we have: x = rsin(θ)cos(φ) y = rsin(θ)sin(φ) z = rcos(θ) Substituting the expression for r into the Cartesian coordinates, we get: x = sin²(θ)sin(φ)sin(θ)cos(φ) y = sin²(θ)sin(φ)sin(θ)sin(φ) z = sin²(θ)sin(φ)cos(θ) Simplifying these equations, we obtain: x = sin³(θ)sin²(φ)cos(φ) y = sin³(θ)sin²(φ)sin(φ) z = sin³(θ)sin(φ)cos(θ) Therefore, the surface described by the equation ρ = sin(θ)sin(φ) is given by the parametric equations: x = sin³(θ)sin²(φ)cos(φ) y = sin³(θ)sin²(φ)sin(φ) z = sin³(θ)sin(φ)cos(θ) To find the point where x = 0, y = 1/2 and z = 0, we substitute these values into the parametric equations: sin³(θ)sin²(φ)cos(φ) = 0 sin³(θ)sin²(φ)sin(φ) = 1/2 sin³(θ)sin(φ)cos(θ) = 0 Solving these equations, we find that the point satisfying the given conditions is (0, 1/2, 0). Answer 4 To determine the surface for which the equation is given, ρ = sinθsinφ, we can use spherical coordinates. In spherical coordinates, ρ represents the distance from the origin to a point, θ is the polar angle (measured from the positive z-axis), and φ is the azimuthal angle (measured from the positive x-axis in the xy-plane). To express the equation ρ = sinθsinφ in terms of spherical coordinates, we substitute the spherical coordinates for ρ, θ, and φ: ρ = sinθsinφ Here, ρ represents the radial distance from the origin to a point. This equation defines a surface in three-dimensional space. To visualize this surface, we can eliminate ρ and express the equation solely in terms of θ and φ: sinθsinφ = 0 Now, let's consider the possible values of θ and φ that satisfy this equation: When sinθ = 0 and sinφ = 0, we have θ = kπ and φ = mπ, where k and m are integers. In this case, the equation becomes 0 = 0, which is true for all values of θ and φ. Therefore, the surface is not well-defined for this case. When sinφ = 0 and sinθ ≠ 0, we have θ = kπ and φ ≠ mπ. In this case, the equation becomes 0 = 0, which is true for all values of θ and φ as long as φ is not an integer multiple of π. This implies that the surface contains all points on the z-axis except for the origin. When sinθ ≠ 0 and sinφ = 0, we have θ ≠ kπ and φ = mπ. In this case, the equation becomes sinθ = 0, which is satisfied when θ = kπ where k is an integer. This implies that the surface contains all points on the xy-plane. In conclusion, the surface defined by the equation ρ = sinθsinφ consists of all points on the z-axis (excluding the origin) and the xy-plane. Question 2 tan(x)sin(x) + sec(x)cos²(x) = sec(x) Answer tan(x) = sin(x) / cos(x) tan(x)sin(x) = sin²(x) / cos(x) sec(x) = 1/cos(x) sec(x)cos²(x) = cos(x) tan(x)sin(x) + sec(x)cos²(x) = sin²(x) / cos(x) + cos(x) = (sin²(x) + cos²(x)) / cos(x) = 1 / cos(x) = sec(x) {proved} as we know sin²(x) + cos²(x) = 1. Question 3 Convert the following angle from degrees to radians: 240° Answer To convert from degrees to radians, multiply by π/180° 240° * π/180° = 240π / 180 = (60 * 4π) / (60 * 3) = 4π/3 Question 4 The terminal side of an angle θ in standard position intersects the unit circle at (5/13, 12/13) What is sin(θ)? Write your answer in simplified, rationalized form. Answer Given a point (x, y) on a unit circle, then: sin(θ) = y From the point, x = 5/13 and y = 12/13 so: sin(θ) = 12/13 Question 5 (sin(a - b)) / (cos a cos b) = tan a - tan b Answer Work using the left side. Use the sine of a difference: sin(x - y) = sinxcos y - cosxsin y sin(a - b) / (cos a cos b) = (sin a cos b - cos a sin b) / (cos a cos b) Separate as: (sin(a - b)) / (cos a cos b) = (sin a cos b / (cos a cos b) ) - (cos a sin b / (cos a cos b)) Cancel common factors: (sin(a - b)) / (cos a cos b) = sin a / cos a - sin b / cos b Use the quotient identity: tan x = sin x / cos x (sin(a - b)) / (cos a cos b) = tan a - tan b