Angle of Elevation and Depression

• Angle of elevation and angle of depression are concepts used in trigonometry to describe the angle between a horizontal line and an observer's line of sight to an object.
• Trigonometric ratios relate the angles of a right triangle to the ratios of its sides.
• These concepts are applied in solving word problems involving heights, distances, and angles.

Angle of Elevation

• Angle of elevation is the angle formed between the horizontal line of sight and the line of sight to an object above the horizontal line.
• It is always measured upwards from the horizontal.
• Imagine a person standing on the ground looking up at the top of a building; the angle of elevation is the angle between the ground (horizontal line) and the person's line of sight to the top of the building.

Angle of Depression

• Angle of depression is the angle formed between the horizontal line of sight and the line of sight to an object below the horizontal line.
• It is always measured downwards from the horizontal.
• Imagine a person standing on top of a cliff looking down at a boat on the sea; the angle of depression is the angle between the horizontal line (at the level of the person) and the person's line of sight to the boat.
• The angle of depression from a point A to a point B is congruent to the angle of elevation from point B to point A because they are alternate interior angles formed by parallel lines (horizontal lines) and a transversal (line of sight).

Six Trigonometric Ratios

• Trigonometric ratios are functions that relate the angles of a right triangle to the ratios of its sides. The primary trigonometric ratios are sine, cosine, and tangent. The reciprocal trigonometric ratios are cosecant, secant, and cotangent.

Sine (sin)

• Sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• sin(θ) = Opposite / Hypotenuse

Cosine (cos)

• Cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• cos(θ) = Adjacent / Hypotenuse

Tangent (tan)

• Tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
• tan(θ) = Opposite / Adjacent

Cosecant (csc)

• Cosecant of an angle is the reciprocal of the sine of that angle.
• csc(θ) = Hypotenuse / Opposite = 1 / sin(θ)

Secant (sec)

• Secant of an angle is the reciprocal of the cosine of that angle.
• sec(θ) = Hypotenuse / Adjacent = 1 / cos(θ)

Cotangent (cot)

• Cotangent of an angle is the reciprocal of the tangent of that angle.
• cot(θ) = Adjacent / Opposite = 1 / tan(θ)

Mnemonic for Trigonometric Ratios

• SOH-CAH-TOA is a mnemonic device used to remember the definitions of the trigonometric ratios:
• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Solving Word Problems Involving Trigonometric Ratios, Angle of Elevation, and Angle of Depression

• Draw a Diagram: Sketch the scenario described in the word problem, labeling all known quantities. Include the right triangle formed by the angles of elevation or depression.
• Identify the Knowns and Unknowns: Determine which sides and angles are given and which need to be found.
• Choose the Appropriate Trigonometric Ratio: Select the trigonometric ratio that relates the known sides and angles to the unknown side or angle.
• Set Up the Equation: Write the equation using the chosen trigonometric ratio, substituting the known values.
• Solve the Equation: Solve the equation to find the value of the unknown.
• Check the Solution: Ensure the solution is reasonable and makes sense in the context of the problem.
• Include Units: Provide the answer with the appropriate units.

Examples of Word Problems

• Finding the Height of a Building: From a point on the ground 50 meters away from the base of a building, the angle of elevation to the top of the building is 60 degrees. Find the height of the building.
  • Draw a diagram: Right triangle with base = 50 m, angle of elevation = 60 degrees, and height (opposite side) as the unknown.
  • Choose ratio: Use tangent (tan) since we have the adjacent side and want to find the opposite side.
  • Equation: tan(60°) = Height / 50
  • Solve: Height = 50 * tan(60°) ≈ 50 * 1.732 ≈ 86.6 meters.
• Finding the Distance to a Boat: From the top of a cliff 100 meters high, the angle of depression to a boat is 30 degrees. Find the distance of the boat from the base of the cliff.
  • Draw a diagram: Right triangle with height = 100 m, angle of depression = 30 degrees (which is also the angle of elevation from the boat to the top of the cliff), and base (adjacent side) as the unknown.
  • Choose ratio: Use tangent (tan) since we have the opposite side and want to find the adjacent side.
  • Equation: tan(30°) = 100 / Distance
  • Solve: Distance = 100 / tan(30°) ≈ 100 / 0.577 ≈ 173.2 meters.
• Finding the Angle of Elevation: A ladder 10 meters long leans against a wall, with its base 6 meters away from the wall. Find the angle of elevation of the ladder.
  • Draw a diagram: Right triangle with hypotenuse = 10 m, adjacent side = 6 m, and angle of elevation as the unknown.
  • Choose ratio: Use cosine (cos) since we have the adjacent side and the hypotenuse.
  • Equation: cos(θ) = 6 / 10 = 0.6
  • Solve: θ = cos⁻¹(0.6) ≈ 53.13 degrees.

Tips for Solving Trigonometry Problems

• Always draw a clear diagram to visualize the problem.
• Label all known and unknown quantities.
• Choose the appropriate trigonometric ratio based on the given information.
• Use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles when the ratios are known.
• Be mindful of units and provide answers with the correct units.
• Check the reasonableness of your answer within the context of the problem.
• Practice solving a variety of problems to build confidence and skill.