The angle of elevation is the angle between the horizontal line of sight and the line of sight to an object above the observer. It is a crucial concept in right triangle trigonometry and the application of inverse trigonometric functions.
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The angle of elevation is always measured from the horizontal line of sight, and it is positive when the object is above the observer.
In right triangle trigonometry, the angle of elevation is used to determine the height of an object or the distance to an object when the other measurements are known.
Inverse trigonometric functions, such as $\arcsin$, $\arccos$, and $\arctan$, are used to find the angle of elevation when the trigonometric ratios are known.
The angle of elevation is a crucial concept in applications such as surveying, navigation, and astronomy, where it is used to measure the height of objects or the distance to celestial bodies.
The angle of elevation is related to the angle of depression, which is the angle between the horizontal line of sight and the line of sight to an object below the observer.
Review Questions
Explain how the angle of elevation is used in right triangle trigonometry to solve problems.
In right triangle trigonometry, the angle of elevation is used to determine the height of an object or the distance to an object when the other measurements are known. For example, if the angle of elevation and the distance to an object are known, the height of the object can be calculated using the trigonometric ratios, such as $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, where $\theta$ is the angle of elevation. Similarly, if the height of an object and the angle of elevation are known, the distance to the object can be calculated using the same trigonometric ratios.
Describe how inverse trigonometric functions are used in the context of the angle of elevation.
Inverse trigonometric functions, such as $\arcsin$, $\arccos$, and $\arctan$, are used to find the angle of elevation when the trigonometric ratios are known. For instance, if the ratio of the opposite side to the adjacent side of a right triangle is known, the angle of elevation can be determined using the $\arctan$ function. This is particularly useful in applications where the angle of elevation needs to be calculated, such as in surveying, navigation, or astronomy, where the height of an object or the distance to a celestial body is measured.
Explain the relationship between the angle of elevation and the angle of depression, and discuss their practical applications.
The angle of elevation and the angle of depression are complementary angles, meaning their sum is 90 degrees. The angle of depression is the angle between the horizontal line of sight and the line of sight to an object below the observer, which is the complement of the angle of elevation. These angles are used in a variety of practical applications, such as surveying, where the angle of elevation is used to measure the height of an object, and the angle of depression is used to measure the depth of a valley or the distance to an object below the observer. In navigation and astronomy, the angle of elevation is used to measure the altitude of celestial bodies, while the angle of depression is used to measure the depth of an object below the horizon.
Related terms
Right Triangle Trigonometry: The study of the relationships between the sides and angles of right triangles, which is used to solve problems involving angles of elevation and depression.
The functions that undo the operations of the basic trigonometric functions, allowing for the determination of an angle given the values of the trigonometric ratios.
The angle between the horizontal line of sight and the line of sight to an object below the observer, which is the complement of the angle of elevation.