Right Triangle Trigonometry Quiz

Right Triangle Trigonometry

• SOH CAH TOA is a helpful acronym for remembering trigonometric ratios:
  • Sine (SOH): Opposite side / Hypotenuse
  • Cosine (CAH): Adjacent side / Hypotenuse
  • Tangent (TOA): Opposite side / Adjacent side

Identifying Sides in a Right Triangle

• The hypotenuse is always the longest side and is opposite the right angle.
• The opposite side is the side across from the reference angle.
• The adjacent side is the side next to the reference angle (not the hypotenuse).
• The reference angle is the angle used to determine the opposite and adjacent sides.

Solving for Unknown Sides

• Scenario 1: If two sides of a right triangle are known, use the Pythagorean theorem to find the third side.
  • Pythagorean Theorem: c² = a² + b² (where c is the hypotenuse and a and b are the legs)
• Scenario 2: If one side and one angle are known, use a trigonometric ratio to find the unknown side.
  • Choose the ratio that includes the known side and the unknown side.

Example: Finding the Length of a Side

• Example: A right triangle with a 49-degree angle, a hypotenuse of 17.6 units, and an unknown side opposite the angle.
  • Use the sine ratio (since we know the hypotenuse and want the opposite side).
  • sin(49°) = opposite / hypotenuse
  • sin(49°) = x / 17.6
  • Solve for x: x = 17.6 * sin(49°) ≈ 13.3 units

Finding an Angle

• Example: A right triangle with sides of 6 and 5 units, and we want to find the angle opposite the 6-unit side.
  • Use the tangent ratio (relating opposite and adjacent sides).
  • tan(x) = opposite / adjacent = 6 / 5
  • Find the angle x using the inverse tangent function (tan⁻¹): x = tan⁻¹(6/5) ≈ 50.2°

Example: Finding the Angle of Elevation

• Example: A 12-meter high telephone pole is supported by a 14-meter wire. Find the angle between the wire and the ground (angle of elevation).
  • Use the sine ratio (knowing opposite and hypotenuse).
  • sin(θ) = opposite / hypotenuse = 12 / 14
  • Find the angle θ using the inverse sine function (sin⁻¹): θ = sin⁻¹(12/14) ≈ 59°

Key Takeaways

• Trigonometric Ratios: Use SOH CAH TOA to remember relationships between sides and angles in right triangles.
• Pythagorean Theorem: Use this theorem to find a missing side when two sides are known.
• Inverse Trigonometric Functions: Use these functions to find an unknown angle when knowing the values of the trigonometric ratios.
• Application: Use trigonometry to solve practical problems involving right triangles (heights, distances, angles).

Trigonometry Applications: Real-World Problems

• Scenario: A large tree is moved, held upright by two guy wires.
• Details:
  • One wire (12 meters) forms a 56-degree angle with the ground.
  • The other wire makes a 48-degree angle with the ground.
  • Both wires are attached 3 meters down from the top of the tree.
• Goal: Calculate the tree's total height, the second wire's length, and the wires' horizontal separation.

Finding the Height of the Tree

• Calculation: The tree's height splits into two parts: the segment from ground to wire attachment point ("y") and the 3-meter segment above.
  • Using the sine function: sin(56°) = y / 12
  • Solving for "y": y = 12 * sin(56°) ≈ 9.95 meters
• Total Height: y + 3 meters = 9.95 meters + 3 meters = 12.95 meters

Determining the Length of the Second Wire

• Calculation: The second wire's length ("r") is found using the triangle with the 48-degree angle and the height "y."
  • sin(48°) = y / r
  • Solving for "r": r = y / sin(48°) = 9.95 meters / sin(48°) ≈ 13.39 meters

Calculating the Horizontal Distance Between the Wires

• Splitting the Distance: The horizontal distance is the sum of two segments, "a" and "b."
  • "a": From tree base to where the first wire touches the ground.
  • "b": From tree base to where the second wire touches the ground.
• Calculating "a" and "b":
  • Using the tangent function: tan(48°) = y / a => a = y / tan(48°) ≈ 8.96 meters
  • Using the tangent function: tan(56°) = y / b => b = y / tan(56°) ≈ 6.71 meters
• Total Horizontal Distance: a + b = 8.96 meters + 6.71 meters ≈ 15.67 meters

Analyzing the Impact of a Broken Wire

• Scenario: One wire breaks; a replacement is needed.
• Change: The new wire is attached 2 meters below the tree top, giving a new height from the ground to attachment (10.95 meters).
• Goal: Determine the new angle the replacement wire makes with the ground.

Finding the New Angle

• Application of Sine: The sine function is used.
  • sin(θ) = 10.95 meters / 13.39 meters
• Using the Inverse Sine: To find the angle (θ), use the inverse sine function (arcsin):
  • θ = arcsin (10.95 meters / 13.39 meters) ≈ 54.9 degrees
• Rounding: Angle is approximately 55 degrees.