Heights and Distances – Problems 1. The upper ¾ th portion of a vertical pole subtends an angle θ such that tanθ = ⅗ at a point in the horizontal plane through its foot and at distance 40m from the foot. Find the possible height of the vertical pole. Solution: Let h be the height Read more about Heights and Distances – Problems[…]
Values of Trigonometric Ratios of Typical Angles – I (i) Value of 7½° θ = 7½° then 2θ = 15° [∵ 1-cos2θ = 2sin²θ and sin2θ = 2sinθcosθ]. . . = (√3 – √2) (√2 – 1) (ii) Value of cot 82½ cot 82½° = cos (90° – 7½°) = tan 7½° = (√3 – Read more about Values of Trigonometric Ratios of Typical Angles – I[…]
Values of Trigonometric Ratios of Typical Angles 1.Value of sin15°, cos15°, sin75°, cos75°, tan15°, tan75°: (i) sin15° sin15° = sin (45° – 30°) = sin45° cos30° – sin30° cos45° . . (ii) cos15° cos15° = cos (45° – 30°) = cos45° cos30° + sin30 sin45° . cos15° = . Also, sin15° = cos75° = -cos105° Read more about Values of Trigonometric Ratios of Typical Angles[…]
Standard Limits – Problems Theorem 1: If n is a rational number and a > 0 then . If n is positive integer and a > 0 then for any a ϵ R, . If n is real number and a > 0 then . If m and n are any real numbers and a Read more about Standard Limits – Problems[…]
Properties of Focal Chord – II Example 4: Semi-latus rectum is the harmonic mean of SP and SQ, where P and Q are the extremities of focal chord. Solution: PQ is the focal chord and . . . Example 5: Circle described on the focal length as diameter touches the tangent at the vertex. Solution: Read more about Properties of Focal Chord – II[…]
Properties of Focal Chord – I Any point on the parabola y² = 4ax can be taken as (at², 2at), where t is parameter and t ϵ R. Any line passing through the focus of the parabola is called the focal chord of the parabola. Example 1: If the chord joining P ≡ (at²₁, 2at₁) Read more about Properties of Focal Chord – I[…]
Equation of the Chord – Ellipse Equation of the chord of the ellipse whose midpoint is (x₁, y₁). Let the slope of the chord be tan. Then any point on the chord at distance r from the point (x₁, y₁) is (x + rcosθ, y + rsinθ). If this point lies on the ellipse, then Read more about Equation of the Chord – Ellipse[…]
Symmetric and Skew Symmetric Matrices Symmetric Matrix: A square matrix A is said to be a symmetric matrix if AT = A. Examples: 1. .. Hence A = AT then A is symmetric matrices. .. Hence A = AT then A is symmetric matrices. 3. If A is a square matrix then show that A + AT, Read more about Symmetric and Skew Symmetric Matrices[…]
Radical Axis and it’s Properties Radical Axis: The radical axis of two circles is the locus of the point which moves such that the lengths of the tangents drawn from it to the two circles are equal. Consider, S ≡ x² + y² + 2gx+ 2fy +c = 0 … (1) and S ≡ x² Read more about Radical Axis and it’s Properties[…]
Direction of Cosines and Direction Ratios Theorem: If l, m, n are the direction cosines of a line L then l² + m² + n² = 1. Proof: l = cosα = x/r m = cosβ = y/r n = cos γ = z/r cos²α + cos²β + cos²γ = 1 (x/r) ² + (y/r) Read more about Direction of Cosines and Direction Ratios[…]