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Nevaeh leans a 22-foot ladder against a wall so that it forms an angle of 64 degrees with the ground. How high up the wall does the ladder reach? Round your answer to the nearest hundreth of a foot if necessary
The ladder reaches approximately 19.77 feet up the wall.
To find how high up the wall the ladder reaches, we can use trigonometry, specifically the sine function.
In this scenario, the ladder represents the hypotenuse of a right triangle, the height of the wall represents the opposite side, and the distance from the base of the wall to the foot of the ladder represents the adjacent side.
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Given that the angle formed by the ladder and the ground is 64 degrees, we’ll use the sine function:
sin(angle) = opposite / hypotenuse
sin(64 degrees) = height / 22 feet
Now, we solve for the height:
height = sin(64 degrees) * 22 feet
Using a calculator:
height ≈ sin(64 degrees) * 22 ≈ 0.8988 * 22 ≈ 19.776 feet
Rounding to the nearest hundredth:
height ≈ 19.78 feet
So, the ladder reaches approximately 19.78 feet up the wall.
Heights and Distances in Trigonometry
In trigonometry, heights and distances refer to the measurement of vertical heights or distances between objects or points. Trigonometry provides tools to calculate these heights and distances using relationships between angles and side lengths in triangles.
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The primary trigonometric functions used to solve problems involving heights and distances are sine, cosine, and tangent. These functions relate the angles of a right triangle to the ratios of its side lengths.
Here’s a brief overview of how these functions are used:
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Sine (sin): In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, sin(θ) = opposite / hypotenuse.
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Cosine (cos): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Mathematically, cos(θ) = adjacent / hypotenuse.
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Tangent (tan): Tangent is the ratio of the length of the side opposite to an angle to the length of the side adjacent to the angle. Mathematically, tan(θ) = opposite / adjacent.
These trigonometric ratios can be used to solve various types of problems involving heights and distances. For example, if you know the length of one side of a right triangle and one of the non-right angles, you can use trigonometric functions to find the lengths of other sides or the heights or distances involved.
Additionally, trigonometric functions can be used in real-world applications such as navigation, engineering, physics, and astronomy to calculate heights and distances between objects or points, given certain angles and side lengths.
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