Trigonometric Ratios and Applications

Page 1: Rig Ratios

• Introduction to trigonometric ratios.

Page 7: Assignment

• Tasks:

  • Complete IXL Q1, Q11.

  • Find the side length using the tangent ratio.

• Due Date: April 5, 2024, at 11:59 PM.

Page 8-9: Sine and Cosine of Complementary Angles

• Key Idea:

  • The sine of an acute angle equals the cosine of its complement.

  • For complementary angles A and B:

    • ( \sin A = \cos(90° - A) = \cos B )

    • ( \sin B = \cos(90° - B) = \cos A )

Page 10: Rewriting Trigonometric Expressions

• Example:

  • Rewrite ( \sin 56° ) in terms of cosine:

    • ( \sin 56° = \cos(90° - 56°) = \cos 34° )

Page 11: Finding Leg Lengths

• Example:

  • Use cosine ratio to find leg lengths:

    • ( \cos 26° = \frac{y}{14} )

    • Solve for ( y ):

      • ( y \approx 12.6 )

Page 12: Identifying Ratios

• Question: Which ratio does not belong?

  • Ratios: ( \sin B, \cos C, \tan B )

Page 13: Sine and Cosine of 45°

• Example:

  • For a 45°-45°-90° triangle:

    • ( \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} \approx 0.7071 )

Page 14-15: Angles of Elevation and Depression

• Definitions:

  • Angle of elevation: Angle formed by a horizontal line and a line of sight up to an object.

  • Angle of depression: Angle formed by a horizontal line and a line of sight down to an object.

Page 16: Modeling Real Life

• Example:

  • Find distance ( x ) to the base of a mountain using angle of depression:

    • ( x = \frac{1200}{\sin 21°} \approx 3349 ) feet.

Page 18: Unit Circle

• Key Idea:

  • Points (0, 0), (x, y), and (x, 0) form a right triangle with leg lengths x and y, hypotenuse length of 1.

Page 19: Using the Unit Circle

• Example:

  • Find trigonometric ratios for angle A using coordinates:

    • ( \sin A = \frac{4}{5}, \cos A = \frac{3}{5}, \tan A = \frac{4}{3} )

Page 21-22: Real-Life Applications

• Example:

  • Find the height of the Ponce de Leon Inlet Lighthouse:

    • Given distance and angle of elevation:

      • ( \tan 59° = \frac{h}{105} )

      • ( h \approx 175 ) feet.

    • Check Reasonableness:

      • Use properties of a 30°-60°-90° triangle for verification.

Page 23: Additional Application

• Task:

  • Measure distance from a lamppost and find its