Trigonometrical Properties
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Questions and Answers

What is the formula for cos A in a triangle ABC?

  • cos A = b^2 + c^2 - a^2 / 2bc (correct)
  • cos A = a^2 - b^2 + c^2 / 2bc
  • cos A = a^2 + b^2 - c^2 / 2bc
  • cos A = b^2 - c^2 - a^2 / 2bc

    • What is the formula for tan B in a triangle ABC where R is the radius of the circumcircle?

  • tan B = R(c - a + b) / 2
  • tan B = R(c + a - b) / 2
  • tan B = R(a + b - c) / 2
  • tan B = R(b + c - a) / 2 (correct)

    • What is the formula for the square of side a in a triangle ABC?

  • a^2 = b^2 + c^2 - 2bc cos A (correct)
  • a^2 = b^2 - c^2 + 2bc cos A
  • a^2 = b^2 + c^2 + 2bc cos A
  • a^2 = c^2 + b^2 + 2bc cos A

    • What is the formula for sin A in a triangle ABC where R is the radius of the circumcircle?

    <p>sin A = a / 2R</p> Signup and view all the answers

    What is the formula for the square of side b in a triangle ABC?

    <p>b^2 = c^2 + a^2 - 2ca cos B</p> Signup and view all the answers

    What is the formula for the area of a triangle, with sides a, b, and c?

    <p>$\sqrt{s(s-a)(s-b)(s-c)}$</p> Signup and view all the answers

    What is the formula for sine of an angle in a triangle ABC?

    <p>$\frac{(s-b)(s-c)}{bc}$</p> Signup and view all the answers

    What is the formula for cosine of an angle in a triangle ABC?

    <p>$\frac{(s-a)(s-c)}{bc}$</p> Signup and view all the answers

    What is the formula for tangent of an angle in a triangle ABC?

    <p>$\frac{(s-c)}{(s-a)(s-b)}$</p> Signup and view all the answers

    What is the sum of the squares of the altitudes of a triangle ABC, in terms of the area of the triangle?

    <p>$\frac{a^2+b^2+c^2}{\Delta}$</p> Signup and view all the answers

    If two sides and the included angle of a triangle are given, what is the formula to find the area of the triangle?

    <p>$\frac{1}{2}bc\sin A$</p> Signup and view all the answers

    If three sides of a triangle are given, what is the formula to find the area of the triangle?

    <p>$\sqrt{s(s-a)(s-b)(s-c)}$</p> Signup and view all the answers

    If three sides and the circum-radius of a triangle are given, what is the formula to find the area of the triangle?

    <p>$\frac{abc}{4R}$</p> Signup and view all the answers

    If $a = 2x$, $b = 2y$, and $C = 120^\circ$ in a triangle, what is the area of the triangle?

    <p>$3xy$</p> Signup and view all the answers

    If $a = 2$, $b = c$, and $\cos A = \cos B = \cos C$ in a triangle, what is the area of the triangle?

    <p>$2$</p> Signup and view all the answers

    What is the point where the perpendicular bisectors of the sides of a triangle intersect?

    <p>The circumcentre</p> Signup and view all the answers

    What is the radius of the circle that can be inscribed within a triangle denoted by?

    <p>r</p> Signup and view all the answers

    What is the formula for the circum-radius of a triangle?

    <p>R = (abc) / (4Δ)</p> Signup and view all the answers

    What passes through the angular points of a triangle?

    <p>Circumcircle</p> Signup and view all the answers

    What is the centre of the inscribed circle of a triangle?

    <p>The incentre</p> Signup and view all the answers

    What is the ratio of the area of a circle to the area of a regular polygon of n sides, where the perimeter of the circle is equal to the perimeter of the polygon?

    <p>$\pi : \tan(\pi/n)$</p> Signup and view all the answers

    If a is the length of each side of a regular polygon of n sides, what is the sum of the radii of the inscribed and circumscribed circles?

    <p>$a\cot(\pi/n) + a\tan(\pi/n)$</p> Signup and view all the answers

    If a is the length of each side of a regular polygon of n sides, what is the radius of the inscribed circle?

    <p>$a\cot(\pi/n)$</p> Signup and view all the answers

    What is the formula for the area of a regular polygon of n sides, where a is the length of each side?

    <p>$\frac{n\pi a^2}{4\tan(\pi/n)}$</p> Signup and view all the answers

    If a is the length of each side of a regular polygon of n sides, what is the radius of the circumscribed circle?

    <p>$a\sin(\pi/n)$</p> Signup and view all the answers

    What is the radius of the inscribed circle of a triangle ABC?

    <p>r = (s - a) tan A/2</p> Signup and view all the answers

    What is the centre of the escribed circle opposite to the angle A in a triangle ABC?

    <p>The point of intersection of the external bisectors of angles B and C</p> Signup and view all the answers

    If r1, r2, and r3 are the radii of the escribed circles opposite to the angles A, B, and C respectively, what is the relationship between them?

    <p>r1 = r2 = r3</p> Signup and view all the answers

    What is the condition for the existence of an escribed circle opposite to the angle A in a triangle ABC?

    <p>AB + AC &gt; BC</p> Signup and view all the answers

    If R is the circumradius of a triangle ABC, what is the relationship between the inradius r and R?

    <p>r = R/2</p> Signup and view all the answers

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