12345 - Math Quiz - Trigonometry Covering: trigonometry, which will be on trigonometric functions and ratios, Pythagorean theorem, and special right triangles (30-60-90 and 45-45-90). 1 / 5 1. A 30-60-90 triangle has a hypotenuse of length 16 cm. What is the length of the side opposite the 60° angle? a) 4√3 cm b) 8√3 cm c) 12√3 cm d) 16√3 cm In a 30-60-90 triangle, the side opposite the 30° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c"), and the side opposite the 60° angle (let's call it "b") is √3 times the length of the side opposite the 30° angle: a = c/2 b = a√3 Now we know that the hypotenuse has a length of 16 cm: a = 16/2 a = 8 cm Now we can find the length of side b: b = 8√3 cm 2 / 5 2. A right triangle has one angle measuring 45° and one leg of length 6√2 cm opposite the 45° angle. Find the length of the hypotenuse. a) 9 cm b) 12 cm c) 15 cm d) 18 cm The right triangle with one angle measuring 45° must have the other acute angle measuring 45° as well (because it's a right triangle). This is a 45-45-90 triangle. In a 45-45-90 triangle, the hypotenuse is √2 times the length of each leg (sides opposite the 45° angles). We are given the length of one leg (opposite a 45° angle) as 6√2 cm. To find the length of the hypotenuse (let's call it "c"), we can use the following equation: c = 6√2√2 cm c = 6(√2)(√2) cm c = 6(2) cm c = 12 cm 3 / 5 3. If tan(x) = 3/4 and x is an angle in a right triangle, find the value of sin(x). a) 2/5 b) 3/5 c) 4/5 d) 5/4 In a right triangle, tan(x) = opposite side length / adjacent side length. In this case, tan(x) = 3/4, so the length of the opposite side is 3 and the length of the adjacent side is 4. We can use the Pythagorean theorem to find the length of the hypotenuse (c): (3 cm)² + (4 cm)² = c² 9 + 16 = c² 25 = c² c = √25 c = 5 cm Now, we can find the value of sin(x) = opposite side length / hypotenuse length: sin(x) = 3/5. 4 / 5 4. Which of the following is NOT a Pythagorean triple? a) (3, 4, 5) b) (5, 12, 13) c) (7, 24, 25) d) (8, 15, 18) A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem, a² + b² = c². First, we will rationalize the first option. (3,4,5): (3 cm)² + (4 cm)² = (5 cm)²?; 9 + 16 = 25; 25 = 25. Next, (5,12,13): (5 cm)² + (12 cm)² = (13 cm)²?; 25 + 144 = 169; 169 = 169. Now for (7,24,25): (7 cm)² + (24 cm)² = (25 cm)²?; 49 + 576 = 625; 625 = 625. Last, (8,15,18): (8 cm)² + (15 cm)² = (18 cm)²?; 64 + 225 = 289; 289 ≠ 324. The Pythagorean triple that doesn't meet the condition is (8, 15, 18). 5 / 5 5. Given that ΔABC is a right triangle with ∠BCA = 60° and AB = 8, where the triangle is inscribed in a circle with its hypotenuse serving as the diameter, determine the area of the circle. a) 27π b) 44.5π c) 64π d) 40√3π Since ΔABC is a right triangle with ∠BCA = 60° and AB = 8, we can find the length of the hypotenuse AC using the Pythagorean theorem:AC² = AB² + BC² AC² = 8² + (8 * tan(60°))² AC² = 64 + 64 * (√3)² AC² = 64 + 64 * 3 AC² = 64 + 192 AC² = 256Now, find the length of the hypotenuse AC:AC = √256 = 16 cmSince the hypotenuse serves as the diameter of the circle, the radius R is half the length of the hypotenuse:R = AC / 2 = 16 / 2 = 8 cmFinally, find the area of the circle using the formula:Area = π * R²In this case, the area of the circle is:Area = π * (8)² = 64π square centimeters Your score is 0% Restart Quiz