Find the values of x and y on the 45-45-90 degree special right triangle. Use exact values, meaning leave the square root in the answer if needed.
In a 45-45-90 triangle, the two legs are congruent and the hypotenuse is the length of one leg times the square root of 2.
Let's call the length of each leg x. Then the length of the hypotenuse is x * √2.
Since the two legs are congruent, we can set up the equation:
x * √2 = x
To solve for x, we divide both sides of the equation by √2:
x = x / √2
Simplifying the right side of the equation, we get:
x = (x * √2) / √2
The √2 on the numerator and denominator cancels out, so we are left with:
x = x / 1
This means that x = x, which is true for any value of x.
Therefore, in a 45-45-90 triangle, the values of x and y can be any positive real numbers.
A plane is descending at a
12°
angle of depression. If the current altitude of the plane is 1,000 feet, find the distance the plane still needs to fly to reach the ground. Round the answer to the nearest foot.(1 point)
We can set up a right triangle to solve this problem, with the vertical leg representing the altitude of the plane, the horizontal leg representing the distance the plane still needs to fly, and the angle of depression being the angle between the hypotenuse and the vertical leg.
We know that the angle of depression is 12° and the altitude of the plane is 1,000 feet. We want to find the distance the plane still needs to fly.
Using trigonometry, we can set up the following equation:
tan(12°) = altitude / distance
Solving for distance, we get:
distance = altitude / tan(12°)
distance = 1000 / tan(12°)
Using a calculator, we find that tan(12°) is approximately 0.21255656167.
Plugging this value into the equation, we can calculate the distance:
distance ≈ 1000 / 0.21255656167
distance ≈ 4708.97 feet
Rounded to the nearest foot, the distance the plane still needs to fly to reach the ground is approximately 4709 feet.
A 20-foot ladder leans against a wall so that the ladder's angle of elevation is
46°
. Find x, the distance from the base of the ladder to the building.(1 point)
We can set up a right triangle to solve this problem, with the ladder representing the hypotenuse, the vertical leg representing the distance from the base of the ladder to the building (x), and the angle of elevation being the angle between the hypotenuse and the horizontal leg.
We know that the angle of elevation is 46° and the length of the ladder is 20 feet. We want to find the distance from the base of the ladder to the building (x).
Using trigonometry, we can set up the following equation:
sin(46°) = opposite / hypotenuse
Solving for x (opposite side), we get:
x = hypotenuse * sin(46°)
x = 20 * sin(46°)
Using a calculator, we find that sin(46°) is approximately 0.7193.
Plugging this value into the equation, we can calculate x:
x = 20 * 0.7193
x ≈ 14.386 feet
Rounded to the nearest foot, the distance from the base of the ladder to the building is approximately 14 feet.
If
c=17
in. and
b=15
in., find the value of
sinθ
. Write your answer as a fraction. You do not need to simplify.
To find the value of sinθ, we need to use the given values of c and b in the equation sinθ = b / c.
Plugging in the values, we get:
sinθ = 15 / 17
Therefore, the value of sinθ is 15/17.
Deshaun is looking up at a flagpole that is 50 feet away from him at an angle of elevation from ground level of
35°
. What is the flagpole's height,
x
, and the distance between Deshaun and the top of the flagpole,
y
?(1 point)
To solve this problem, we can set up a right triangle with the height of the flagpole being the vertical leg (x), the distance between Deshaun and the top of the flagpole being the horizontal leg (y), and the angle of elevation being the angle between the hypotenuse and the horizontal leg.
We know that the angle of elevation is 35° and the distance between Deshaun and the flagpole is 50 feet. We want to find the height of the flagpole (x) and the distance between Deshaun and the top of the flagpole (y).
Using trigonometry, we can set up the following equations:
sin(35°) = x / 50 (since sinθ = opposite / hypotenuse)
cos(35°) = y / 50 (since cosθ = adjacent / hypotenuse)
To solve for x, we can rewrite the equation as:
x = sin(35°) * 50
Using a calculator, we find that sin(35°) is approximately 0.5736.
Plugging this value into the equation, we can calculate x:
x ≈ 0.5736 * 50
x ≈ 28.68 feet
Therefore, the height of the flagpole is approximately 28.68 feet.
To solve for y, we can rewrite the equation as:
y = cos(35°) * 50
Using a calculator, we find that cos(35°) is approximately 0.8192.
Plugging this value into the equation, we can calculate y:
y ≈ 0.8192 * 50
y ≈ 40.96 feet
Therefore, the distance between Deshaun and the top of the flagpole is approximately 40.96 feet.