Trigonometry: Angle of Elevation and Depression
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Questions and Answers

What is the angle of elevation defined as?

  • The angle formed at the intersection of two horizontal lines.
  • The angle created by extending a horizontal line from an object to a point above it.
  • The angle between the ground and a vertical line extending to an object below the horizontal line.
  • The angle between the horizontal line and the line of sight to an object above the horizontal line. (correct)

    • Which case requires the knowledge of both sides and angles to solve for unknown lengths or angles?

  • Case 1: Given an angle and a side
  • The Law of Sines
  • The Law of Cosines
  • Case 2: Given two sides (correct)

    • In the context of trigonometry, how is the Law of Sines expressed?

  • $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (correct)
  • $\frac{a}{A} + \frac{b}{B} + \frac{c}{C} = 1$
  • $a^2 + b^2 = c^2$
  • $\tan A = \frac{opposite}{adjacent}$

    • How is the Law of Cosines formulated?

    <p>$c^2 = a^2 + b^2 - 2ab \cos C$</p> Signup and view all the answers

    What is the relationship between angle of depression and angle of elevation?

    <p>They are equal when measured from the same horizontal line.</p> Signup and view all the answers

    Study Notes

    Angle of Elevation and Angle of Depression

    • The angle of elevation is the angle formed by a horizontal line of sight and the line of sight to an object that is above the horizontal.
    • The angle of depression is the angle formed by a horizontal line of sight and the line of sight to an object that is below the horizontal.
    • These angles are crucial in solving problems involving heights and distances, particularly in surveying and navigation.

    Case 1: Given an Angle and a Side

    • In this case, you are provided with a trigonometric function (sine, cosine, tangent) relating to the angle of elevation or depression and the side of a right triangle.
    • To solve for an unknown side or angle, use the trigonometric ratios and substitute the known values into the equation.
    • Example: If given the angle of elevation to the top of a building (a) and the horizontal distance from the observer to the base of the building (b), you can use the tangent function to find the height of the building (h): tan(a) = h/b => h = b * tan(a).

    Case 2: Given Two Sides

    • Scenario 1 - Right Triangles: If you are given two sides of a right triangle, you can use the Pythagorean theorem (a² + b² = c²) to find the third side, and then use trigonometric ratios to find the angles of elevation or depression.

    • Scenario 2 - Oblique Triangles: If you are given two sides of a non-right triangle, you need the Law of Sines or Law of Cosines to find angles and remaining sides.

    Law of Sines

    • The Law of Sines states the following relationship for any triangle ABC: a / sin A = b / sin B = c / sin C where a, b, and c are the side lengths opposite to the angles A, B, and C, respectively.
    • Used to solve oblique triangles when you know two angles and one side (ASA or AAS) or two sides and the angle opposite one of them (SSA).

    Law of Cosines

    • The Law of Cosines states the following relationships for any triangle ABC: a² = b² + c² - 2bc * cos A b² = a² + c² - 2ac * cos B c² = a² + b² - 2ab * cos C

    • Used to solve oblique triangles when you know two sides and the included angle (SAS) or three sides (SSS).

    • Example: To find the third side of a triangle given two sides and the included angle (SAS), use the Law of Cosines. To find the third side given three sides (SSS), calculate the angle (either A, B, or C) first and then apply the Law of Cosines or Law of Sines.

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    Description

    This quiz focuses on the concepts of angle of elevation and angle of depression, essential in solving trigonometric problems related to heights and distances. You will apply trigonometric ratios in various scenarios to find unknown sides or angles in right triangles.

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