Overview
This lecture covers the fundamentals of right triangle trigonometry, explains the six basic trigonometric functions, shows how to solve for missing sides and angles, and introduces special right triangles and calculation techniques.
Basics of Right Triangle Trigonometry
- A right triangle has one right angle and three sides: opposite, adjacent, and hypotenuse.
- The Pythagorean theorem: ( a^2 + b^2 = c^2 ) applies to right triangles (a & b are legs, c is hypotenuse).
Trigonometric Functions & SOHCAHTOA
- SOHCAHTOA is a mnemonic for trigonometric ratios:
- Sine (( \sin \theta )) = Opposite / Hypotenuse
- Cosine (( \cos \theta )) = Adjacent / Hypotenuse
- Tangent (( \tan \theta )) = Opposite / Adjacent
- Cosecant = 1 / Sine = Hypotenuse / Opposite
- Secant = 1 / Cosine = Hypotenuse / Adjacent
- Cotangent = 1 / Tangent = Adjacent / Opposite
Special Right Triangles
- Common Pythagorean triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples.
- Knowing triplets allows quick identification of missing sides.
Calculating Trig Functions from Side Lengths
- Example: For a triangle with sides 3, 4, and 5:
- ( \sin \theta = 4/5 ), ( \cos \theta = 3/5 ), ( \tan \theta = 4/3 )
- ( \csc \theta = 5/4 ), ( \sec \theta = 5/3 ), ( \cot \theta = 3/4 )
- Repeat for triangles with different sides by identifying which sides are opposite, adjacent, or hypotenuse.
Solving for Side Lengths
- Use the Pythagorean theorem for missing sides.
- For non-special triangles, use trigonometric functions with known angle and side to solve for unknowns.
Solving for Missing Angles
- If two sides are known, use the appropriate trig ratio and its inverse function (e.g., ( \theta = \arctan(\text{opposite}/\text{adjacent}) )).
- Use a calculator in degree mode to determine angle measures.
Using the Calculator
- Double-check calculator is set to degrees.
- Use functions (\sin^{-1}), (\cos^{-1}), or (\tan^{-1}) for finding angles.
Additional Topics (Course Overview)
- Course covers angles, radians, unit circle, all six trig functions, special right triangles, angle of elevation/depression, general trig functions, graphing, inverse functions, and verifying identities.
- Future content includes law of sines, law of cosines, and polar coordinates.
Key Terms & Definitions
- Opposite Side — The side across from the angle of interest.
- Adjacent Side — The side next to the angle of interest (not the hypotenuse).
- Hypotenuse — The longest side, opposite the right angle.
- SOHCAHTOA — Mnemonic for remembering trig ratios.
- Pythagorean Triplet — Set of integers that satisfy ( a^2 + b^2 = c^2 ).
Action Items / Next Steps
- Practice identifying sides and applying SOHCAHTOA to solve for missing sides/angles.
- Review and memorize common Pythagorean triplets.
- Complete assigned practice problems using trig functions and a calculator in degree mode.