Trigonometry For Beginners!

Introduction to Right Triangle Trigonometry

• Focus on the expression SOA, which is core to understanding trigonometry.

Key Components of a Right Triangle

• Angle Theta (θ): The angle for which we perform trigonometric calculations.

• Sides of the Triangle:

  • Opposite Side: The side opposite the angle Theta.

  • Adjacent Side: The side next to the angle Theta, excluding the hypotenuse.

  • Hypotenuse: The longest side, opposite the right angle.

Pythagorean Theorem

• The relationship between sides of a right triangle: A² + B² = C², where C is hypotenuse.

Trigonometric Functions using SOA

• Sine (SOA):

  • Formula: Sin(θ) = Opposite / Hypotenuse

• Cosine (CAH):

  • Formula: Cos(θ) = Adjacent / Hypotenuse

• Tangent (TOA):

  • Formula: Tan(θ) = Opposite / Adjacent

Reciprocal Trigonometric Functions

• Cosecant:

  • Csc(θ) = 1 / Sin(θ) = Hypotenuse / Opposite

• Secant:

  • Sec(θ) = 1 / Cos(θ) = Hypotenuse / Adjacent

• Cotangent:

  • Cot(θ) = 1 / Tan(θ) = Adjacent / Opposite

Example Problem 1

• Given: Triangle sides of 3 and 4, compute the hypotenuse (C).

  • Pythagorean theorem: 3² + 4² = C²

  • Calculation: C = √(9 + 16) = √25 = 5

• Calculate Trigonometric Functions:

  • Sin(θ) = Opposite/Hypotenuse = 4/5

  • Cos(θ) = Adjacent/Hypotenuse = 3/5

  • Tan(θ) = Opposite/Adjacent = 4/3

  • Csc(θ) = 5/4, Sec(θ) = 5/3, Cot(θ) = 3/4

Special Right Triangles

• Notable side lengths: 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

• Multiples of these ratios also work (e.g., 6-8-10, 9-12-15).

Example Problem 2

• Given: Sides of 8 and 17:

  • Hypotenuse calculation: 8² + B² = 17² → B = 15

• Trigonometric Values:

  • Sin(θ) = 15/17

  • Cos(θ) = 8/17

  • Tan(θ) = 15/8

  • Csc(θ) = 17/15, Sec(θ) = 17/8, Cot(θ) = 8/15

Example Problem 3

• Given: Hypotenuse 25 and side 15:

  • Identify missing side using ratios: Missing side = 20

• Trigonometric Values:

  • Sin(θ) = 20/25 = 4/5

  • Cos(θ) = 15/25 = 3/5

  • Tan(θ) = 20/15 = 4/3

  • Csc(θ) = 5/4, Sec(θ) = 5/3, Cot(θ) = 3/4

Finding Missing Sides and Angles

• Use trigonometric functions based on given values (opposite, adjacent, hypotenuse) to solve for missing sides or angles.

• Example: For angle 38° and adjacent side of 42, use Tan(38°) = x/42 to solve for x.

• Calculating Specific Values:

  • Transition from triangle sides to angles using inverse functions (e.g., θ = tan⁻¹(5/4)) for calculations.

Advanced Applications and Further Learning

• Mention of advanced properties and sections available through Trigonometry course on Udemy.

  • Coverage includes unit circle, right triangle properties, angle elevation problems, identities, and more.

  • Links to mathematical formulations and reciprocal trigonometric functions.

Conclusion

• Reinforce the importance of these concepts for effective problem-solving in trigonometry.