Explain how a trigonometric ratio can be used to find a measurement indirectly.
Say you want to find the height of a building.
If you know the angle up to the top, and how far away you are, you can use the tangent function to determine the height without having to measure it directly.
Similarly, you can determine the diameter of the earth or the distance to stars. To see how, fire up a google search.
Trigonometric ratios, such as sine, cosine, and tangent, can be used to find measurements indirectly through the use of right triangles.
To understand how this works, let's consider an example:
Suppose you want to find the height of a tall building, but you are unable to measure it directly. However, you have a measuring tape, and you can measure the distance from the base of the building to a point directly in front of it. Let's call this distance "d."
To find the height of the building, you can use the trigonometric ratio tangent. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
Here's how you can find the height:
1. Stand at a point directly in front of the building and measure the distance from the base of the building to your location. Let's call this distance "d."
2. Now measure the angle between your line of sight and the ground. You can use a clinometer or a smartphone app to measure angles.
3. Let's call the angle you measured "θ."
4. Draw a right triangle where the base of the triangle represents your distance "d" and the height of the building represents the height you want to find.
5. Your distance "d" is the adjacent side of the triangle, and the height of the building is the opposite side. The angle θ is the angle between these two sides.
6. Now, apply the definition of tangent. Tangent θ is equal to the opposite side divided by the adjacent side. So, in this case, the tangent θ is equal to the height of the building divided by the distance "d," which can be written as tan(θ) = height/d.
7. Rearrange the equation to solve for the height of the building. Multiply both sides by "d" to isolate the height, so the equation becomes height = d * tan(θ).
8. Finally, substitute the values of "d" and θ into the equation, and calculate the height using a scientific calculator or trigonometric tables.
By applying this method, you can indirectly find the height of the building using trigonometric ratios and basic principles of right triangles.
Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a right triangle to the ratios of its sides. These ratios are widely used in geometry and trigonometry to solve various real-life problems, such as finding unknown measurements indirectly.
To understand how a trigonometric ratio can be used to find a measurement indirectly, let's consider the example of finding the height of a tall object, like a tree or a building. Suppose you are standing some distance away from the object and want to determine its height, but you cannot directly measure it.
Here's how you can use a trigonometric ratio, like tangent, to find the height indirectly:
1. Choose a reference point: Select a point from where you can see the top of the object, such as your eye level or any elevated spot.
2. Measure the distance: Determine the distance between your reference point and the base of the object. You can use a measuring tape or other distance measuring tools for this step.
3. Measure the angle: Use a clinometer or any other tool that allows you to measure angles. Point it towards the top of the object and measure the angle of elevation. This angle represents the angle between your line of sight and the horizontal ground.
4. Apply the trigonometric ratio: In a right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. In our case, the opposite side is the height of the object, and the adjacent side is the distance between your reference point and the object's base.
5. Calculate the height: Using the tangent ratio, you can set up an equation and solve for the height of the object. Substitute the known values (distance and angle) into the tangent function and solve for the height.
By using trigonometric ratios, you are able to indirectly find the height of the object without directly measuring it. These ratios provide a mathematical relationship between angles and sides of a triangle, enabling us to solve problems where direct measurements are not feasible.