Trigonometry Study Notes

Trigonometry Study Notes

Basic Trigonometric Ratios

  • Express each ratio as a fraction and decimal to the nearest hundredth:

    1. ratio A:

    • Fraction: \frac{16}{20}

    • Decimal: -0.80

    1. tan C:

    • Fraction: \frac{12}{16}

    • Decimal: 0.75

    1. cos A:

    • Fraction: \frac{12}{20}

    • Decimal: -0.60

    1. tan 4:

    • Fraction: \frac{16}{12}

    • Decimal: 1.33

    1. cos C:

    • Fraction: \frac{16}{20}

    • Decimal: -0.30

    1. sin C:

    • Fraction: Used special right triangles to express:

    • Decimal: Detailed below

Special Right Triangles and Trigonometric Ratios

Using Special Right Triangles

  • tan 60°:

    • Evaluation: rac{\sqrt{3}}{1} approximately = 1.73

  • cos 30°:

    • Evaluation: \frac{\sqrt{3}}{2} approximately = 0.87

  • sin C for specified values like 60° or 30° gives decimal approximations based on trigonometric functions of specific angles.

    • Example: sin 60° = √3/2 = 0.866

Problems Involving Tangents and Cosines

  • For triangle sides:

    • Triangle with sides 3, 4, 5 cm:

      • Tangent of greater acute angle = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} = 1.33

    • Cosine of smaller acute angle in triangle with sides 10, 24, 26 inches:

      • Calculation = \frac{10}{26} = 0.3846, rounding to ≈ 0.39

Applications of Trigonometry

  1. The springboard in gymnastics with a 14.5° angle to a base:

    • Calculation returns approximately 16.93 inches for length of springboard.

  2. Construction of bike ramp with a length of 3 feet and 20° angle:

    • Height needs to be calculated using: h = 3 * sin(20) leads to approximately 24 inches tall.

  3. Determining the height of a vertical pole supported by a guy wire involves basic trigonometric calculations to find the height using the known angle and wire length, yielding an answer around 1.2 feet.

Solving Right Triangles in Various Situations

Right Triangle Solutions

  • Use Pythagorean Theorem and Trigonometric Ratios

    • For calculations of unknown sides, rounding measurements to nearest hundreds and angles to the nearest degree:

    • Example: If one side is 12 and another is 5, hypotenuse = \sqrt{12^2 + 5^2} = 13.

Calculating Angles

  • Calculate angle measures using inverse trigonometric functions:

    • cos(0.87) yields approximately 29.54°

    • angle for sin(0.85) gives close to 58°

    • General formula: \text{If sin A = x, then A = sin^{-1}(x) }

Special Triangles

  • 45-45-90 triangles and 30-60-90 triangles have specific ratio relationships:

    • 45-45-90 Triangle:

    • Legs = 1, Hypotenuse = \sqrt{2}

    • 30-60-90 Triangle:

    • Hypotenuse = 2 * shorter leg, longer leg = \sqrt{3} * shorter leg

Review of Pythagorean Theorem and Converse

  • Pythagorean theorem states a^2 + b^2 = c^2

  • Converse: If c^2 = a^2 + b^2, triangle is right.

  • Use to classify triangles as acute or obtuse based on relationships of side lengths.

  • Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25).

Practical Applications of Trigonometry in Real World

  • Examples dealing with heights of buildings, slopes, and ramps, often converting between angle measures and lengths using trigonometric ratios and inverse functions, highlighting the importance in real-world applications.