geometry

Geometry Notes: Special Right Triangles and Trigonometry

Pythagorean Theorem

• Used to find the missing side of a right triangle.

• Definitions:

  • Legs (a and b): The two sides of a right triangle that form the right angle.

  • Hypotenuse (c): The side opposite the right angle, the longest side of a right triangle.

• Key Formula: For any right triangle: $a^2 + b^2 = c^2$

Examples and Applications

1. Find the value of x:

   • Example: $78^2 + 13^2 = x^2$

   • Calculation: 64 + 169 = 233

   • Answer: $x = 15.3$

2. Example with missing leg:

   • Given: $9^2 + b^2 = c^2$

   • Find: $x$ where $222 + 272 = x^2$

   • Calculate: $484 + 729 = x^2$

   • Thus, $\sqrt{1213} = x$

3. Another missing side example:

   • $9^2 + b^2 = 12$

   • $11^2 + x^2 = 242$

   • Result is $x = 21.3$.

Other Applications of the Pythagorean Theorem

• Ladder Problem: A roofer leaned a 16-foot ladder against a house with the base 5 feet from the house.

  • Setup: $x^2 + 5^2 = 16^2$.

  • Calculation: $x^2 + 25 = 256$, thus $X = 15.3$.

• Rectangular Deck Problem: Dimensions are 10 feet by 23 feet, find the diagonal.

• Jogging Distance Problem: Ashley's jog of 3.4 miles east and 5.7 miles south is calculated to find total distance from start.

• Support Wire: A 31-foot wire attached from a 25-foot pole uses Pythagorean Theorem to find the distance from the pole's base.

Special Right Triangles

45°-45°-90° Triangle

• Properties: Two legs are of equal length.

  • If leg = x, then hypotenuse = $x\sqrt{2}$.

  • Example Values: If leg = 8, hypotenuse = $8\sqrt{2}$.

30°-60°-90° Triangle

• Properties: The shorter leg (y) is opposite the 30° angle, longer leg = $2y$, hypotenuse = $2y$

  • Example Calculation: If the shorter leg = 5, then longer leg = $5\sqrt{3}$, hypotenuse = 10.

Finding Variables in Special Right Triangles

• Identify angles and apply properties to determine lengths of missing legs and hypotenuses.

  • Calculations:

  1. If shorter leg (y) = 6, then longer leg = $6\sqrt{3}$ and hypotenuse = 12.

  2. If longer leg = 4√3, find y using opposite relationships.

Trigonometric Ratios

Definitions

• Sine (sin): Ratio of the opposite side over the hypotenuse.

  • $ext{sin A} = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine (cos): Ratio of the adjacent side over the hypotenuse.

  • $ext{cos A} = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent (tan): Ratio of the opposite side over the adjacent side.

  • $(\text{tan A} = \frac{\text{opposite}}{\text{adjacent}})$

Example Problems

1. Find value of trigonometric ratios for a triangle with sides 5, 12, 13:

   • $sin A = \frac{5}{13}, cos A = \frac{12}{13}, tan A = \frac{5}{12}$.

2. Trigonometric functions problem: Given a ladder leaning against a wall, use sine, cosine, or tangent to find missing lengths based on angles.

   • Example: If the ladder is 12 feet at an angle of 68° with the ground, determine the distance from the base to the wall using tangent.

Angle of Elevation and Depression

Angle of Elevation

• Formed by the observer's line of sight looking up from the horizontal line.

  • Example: Casey sights the top of the lighthouse at an angle of 58°, 6 feet tall and stands how far from the base?

Angle of Depression

• Formed by the observer's line of sight looking down from the horizontal line.

  • Example: A lifeguard looks down at a swimmer at 18° angle. If the guard is 8 feet high, determine the distance to the swimmer using trigonometry.