1.2 Right Triangle Trigonometry

Page 1: Introduction to Right Triangle Trigonometry

Lesson Objectives

• Introduction to the relationship between sides and angles in right triangles.

• Goals include:

  1. Setup trigonometric ratios from right triangles.

  2. Find missing sides using trig ratios and Pythagorean Theorem.

  3. Solve for missing sides and angles using trig ratios and inverse trig ratios.

  4. Establish basic trigonometric identities.

Trigonometric Ratios

• For a right triangle with angle 𝜃 in quadrant I:

  • Six trigonometric functions can be defined:

    • sin 𝜃 = Opposite/Hypotenuse

    • cos 𝜃 = Adjacent/Hypotenuse

    • tan 𝜃 = Opposite/Adjacent

    • csc 𝜃 = 1/sin 𝜃

    • sec 𝜃 = 1/cos 𝜃

    • cot 𝜃 = 1/tan 𝜃

• To find missing sides, use the Pythagorean Theorem:

  • A triangle is right if and only if:

    • 𝑎² + 𝑏² = 𝑐²

Page 2: Problem-Solving with Trig Functions

Directions

• Find the missing side and all trigonometric functions. Use inverse trig functions to determine missing angles.

Example Problems

• Ex. 1: cot 𝜃 = 1

• Ex. 2: sec 𝜃 = 5

• Additional examples provided with inappropriate symbols inconsistent with standard expressions.

Page 3: Applying Trig Ratios

Tasks

1. Given sin 𝜃 = 4/5, find:

   • a. cos² 𝜃 ∙ tan 𝜃

   • b. sin 𝜃 + csc 𝜃

2. Given tan 𝜃 = 4/3, find:

   • a. sin 𝜃 − 2 csc² 𝜃

   • b. sec 𝜃 + 5 csc 𝜃

3. Given csc 𝜃 = 6/5, find:

   • a. sec² 𝜃 ∙ tan 𝜃

   • b. 2 sin 𝜃 − cos² 𝜃

Page 4: Complementary Angles Theorem

Theorem Overview

• In a right triangle: ∠𝐴 + ∠𝐵 = 90°

• The relationships between angles indicate sine and cosine, tangent and cotangent, as well as secant and cosecant are cofunctions of each other.

Function Relationships

• Cofunctions of complementary angles are equal:

  • sin B = cos A

  • csc B = sec A

  • tan B = cot A

Page 5: General Identities

Important Identities

1. Reciprocal Identities

2. Quotient Identities

3. Pythagorean Identities

Cofunction Identities

• Extensions of the Complementary Angles Theorem:

  • sin 𝜃 = cos(90° − 𝜃)

  • cos 𝜃 = sin(90° − 𝜃)

  • tan 𝜃 = cot(90° − 𝜃)

  • Other related identities using radians are also noted.

Example Problems in Right Triangle Trigonometry

1. Finding Missing Sides Using Trig RatiosGiven that ( \sin \theta = \frac{3}{5} ), find the lengths of the adjacent and hypotenuse sides if the opposite side is 3.

2. Solving for AnglesIf ( \tan \theta = 2 ), find the measure of angl( \theta ) using the inverse trig function.

3. Applying the Pythagorean TheoremIn a right triangle, if one leg is 6 and the other leg is 8, find the hypotenuse using the Pythagorean theorem.

4. Using Reciprocal IdentitiesIf ( \sec \theta = 4 ), find all the other trigonometric functions for angle ( \theta ).

5. Cofunction IdentitiesIf ( \sin A = \frac{5}{13} ), find ( \cos (90° - A) ).

6. Calculating using Multiple RatiosGiven ( \csc \theta = 2 ), determine ( \tan^2 \theta + 1 ).

These problems illustrate various applications of trigonometric functions and identities in solving for sides and angles in right triangles.