Chapter 9: Right Triangles And Trigonometry

The correct answer is "Opposite/Hypotenuse". In trigonometry, the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

The definition of "tangent" refers to the ratio of the length of the side opposite an acute angle in a right triangle to the length of the side adjacent to that angle. In other words, the tangent of an angle is equal to the length of the side opposite the angle divided by the length of the side adjacent to the angle.

The definition of "cosine" is the ratio of the length of the side adjacent to an acute angle in a right triangle to the length of the hypotenuse.

In a right triangle, the side opposite the right angle is called the hypotenuse. Using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can find the length of AB. In this case, AC is one of the legs and BC is the other leg. So, AC^2 + BC^2 = AB^2. Plugging in the given values, we get 4^2 + 3^2 = AB^2. Simplifying, we get 16 + 9 = AB^2. Therefore, AB^2 = 25. Taking the square root of both sides, we find that AB = 5.

In a 45°-45°-90° triangle, the two legs are congruent, meaning they have the same length. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the two legs. In this case, since the two legs are congruent, we can represent their length as "x". Therefore, the length of the hypotenuse is "x√2". So, the ratio of the length of the hypotenuse to the length of a side is "√2:1".

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Using the Pythagorean theorem, we can calculate the length of the second leg. Given that one leg is 4 meters and the hypotenuse is 8 meters, we can use the formula: (4^2) + (x^2) = (8^2). Solving for x, we find that the approximate length of the second leg is 7 meters.

The cosine of angle A can be found using the adjacent side (BC) and the hypotenuse (AB) of the triangle. In this case, BC is 8 and AB is 17. The cosine of angle A is equal to the adjacent side divided by the hypotenuse, which gives us 8/17.

To calculate the length of the ramp, we can use trigonometry. The length of the ramp is the hypotenuse of a right triangle, with the height of the step being the opposite side and the pitch angle being the angle. We can use the sine function to find the length of the ramp. Sin(30°) = opposite/hypotenuse. Rearranging the formula, we get hypotenuse = opposite/sin(30°). Plugging in the values, we get hypotenuse = 25/sin(30°) = 50 inches. Therefore, the length of the ramp will be 50 inches.

The length of the diagonal of a square can be found using the Pythagorean theorem, which states that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the two sides. In this case, the side length of the square is 7 times the square root of 2. So, the length of the diagonal can be found by squaring 7 times the square root of 2 and taking the square root of the result. This simplifies to 7 times the square root of 2. Therefore, the length of the diagonal is 14.

In a triangle, the sum of the squares of the two shorter sides should be greater than the square of the longest side for the triangle to be acute. In this case, 7^2 + 6^2 = 49 + 36 = 85 which is less than 9^2 = 81. Therefore, the triangle with side lengths 7, 6, and 9 is acute.

The ladder forms a right triangle with the side of the house and the ground. The ladder acts as the hypotenuse of the triangle, the base is 8 feet, and the height is unknown. By using the Pythagorean theorem, we can solve for the height. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is 16 feet (the length of the ladder), the base is 8 feet, and the height is unknown. By rearranging the equation and solving for the height, we find that the height is 8 times the square root of 3 feet.

not-available-via-ai

In a 30°-60°-90° triangle, the ratio of the length of the hypotenuse to the length of the shorter side is square root of 3:1. This is because in a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter side, and the length of the longer side (opposite the 60° angle) is equal to the length of the shorter side multiplied by square root of 3. Therefore, the ratio of the length of the hypotenuse to the length of the shorter side is square root of 3:1.

The lengths of the sides of a right triangle must satisfy the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In this case, if we square the lengths of the sides, we find that 7^2 + 24^2 = 49 + 576 = 625, which is equal to 25^2. Therefore, 7, 24, 25 could be the lengths of the sides of a patio in the shape of a right triangle.

The angle of depression is the angle formed between a horizontal line and the line of sight from an observer to a point below the observer. In this case, the angle of depression of the boat is 27°. Since the lighthouse is 210 feet high and was built at sea level, the distance from the top of the lighthouse to the foot of the lighthouse is also 210 feet. We can use trigonometry to find the distance from the boat to the foot of the lighthouse. Using the tangent function, we can set up the equation tan(27°) = 210/d, where d is the distance we want to find. Solving for d, we get d = 210/tan(27°) ≈ 206.9 feet. Therefore, the distance from the boat to the foot of the lighthouse is approximately 207 feet.