Geometry: Pythagorean Theorem and Trigonometry

Yes, it is a right-angled triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, 13² = 5² + 12², which satisfies the theorem.

Sin θ = Opposite / Hypotenuse = 8 / 10 = 4/5.

Let the side opposite the 60-degree angle be 'x'. Using the sine rule, a/sin A = b/sin B, we get 7/sin 40° = x/sin 60°. Solving for x, we get x = (7 * sin 60°) / sin 40°.

<p>Let the third side be 'c'. Using the cosine rule, c² = a² + b² - 2ab cos C, we get c² = 6² + 8² - 2 * 6 * 8 * cos 30°. Solving for c, we get c = √(6² + 8² - 2 * 6 * 8 * cos 30°).</p> Signup and view all the answers

<p>Area = 1/2 * base * height = 1/2 * 10 * 5 = 25 square units.</p> Signup and view all the answers

<p>First, calculate the semi-perimeter: s = (a + b + c) / 2 = (4 + 5 + 6) / 2 = 7.5. Then, using Heron's formula: Area = √[s(s-a)(s-b)(s-c)] = √[7.5(7.5-4)(7.5-5)(7.5-6)] = √(7.5 * 3.5 * 2.5 * 1.5) = √99.22 ≈ 9.96 square units.</p> Signup and view all the answers

<p>SOH CAH TOA is a mnemonic device used to remember the trigonometric ratios: sine (SOH), cosine (CAH), and tangent (TOA). It helps us understand that each ratio relates a specific angle to the lengths of two sides in a right triangle. Sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.</p> Signup and view all the answers

<p>The sine rule is used when you know two angles and one side, or two sides and an angle opposite one of them. It relates the ratio of a side's length to the sine of its opposite angle. The cosine rule is used when you know three sides or two sides and the included angle. It relates the square of one side to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.</p> Signup and view all the answers

In a right triangle, a² + b² = c², where c is the hypotenuse.

The longest side of a right triangle, opposite the right angle.

A mnemonic for trig ratios: Sin = Opp/Hyp; Cos = Adj/Hyp; Tan = Opp/Adj.

In any triangle, a/sin A = b/sin B = c/sin C for finding sides/angles.

a² = b² + c² - 2bc cos A, used for non-right triangles with known sides or angles.

Measure of a triangle's surface, often calculated as 1/2 × base × height.

Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter of a triangle.

Half of the triangle's perimeter: s = (a+b+c)/2.