Trigonometric Functions and Their Applications

Solving Right Triangles

Solving a right triangle involves finding all unknown side lengths and angle measures, given some initial information. This process uses the fundamental trigonometric ratios: sine, cosine, and tangent.

• Right Triangle Definition: A triangle with one angle equal to 90°.

• Trigonometric Ratios:

  • Sine:

  • Cosine:

  • Tangent:

• Solving Steps:

  1. Identify known sides and angles.

  2. Use trigonometric ratios to find missing sides or angles.

  3. Apply the Pythagorean Theorem if two sides are known:

• Example: Given a right triangle with one leg of length 3 and hypotenuse 5, find the other leg and the non-right angles.

Applications Involving Right Triangles and Trigonometric Functions

Trigonometric functions are widely used to solve real-world problems involving heights, distances, and angles.

• Common Applications: Navigation, construction, physics (projectile motion), and engineering.

• Problem-Solving Steps:

  1. Draw a diagram representing the situation.

  2. Label known and unknown quantities.

  3. Set up equations using trigonometric ratios.

  4. Solve for the unknowns.

• Example: Finding the height of a building using the angle of elevation and a measured distance from the base.

Points on Terminal Sides and Trigonometric Function Values

Angles in standard position can be represented by points on their terminal sides. The coordinates of these points are used to define trigonometric functions for any angle.

• Standard Position: An angle with its vertex at the origin and initial side along the positive x-axis.

• Trigonometric Functions from Coordinates: For a point on the terminal side and :

• Example: If a point (3, 4) lies on the terminal side, , so , , .

Finding All Trigonometric Function Values Given One Value and Quadrant

Given one trigonometric function value and the quadrant of the angle, the other five function values can be determined using identities and sign conventions.

• Reciprocal and Pythagorean Identities:

• Sign of Functions by Quadrant:

  • Quadrant I: All positive

  • Quadrant II: Sine positive

  • Quadrant III: Tangent positive

  • Quadrant IV: Cosine positive

• Example: If and is in Quadrant II, , , etc.

Linear Speed and Angular Speed

Linear speed and angular speed describe the motion of objects along circular paths.

• Linear Speed (): The rate at which an object moves along a path: , where is arc length and is time.

• Angular Speed (): The rate at which an angle changes: , where is in radians.

• Relationship: , where is the radius.

• Example: A wheel of radius 0.5 m rotates at radians per second. Linear speed at the rim: m/s.

Reflections on the Unit Circle

Points on the unit circle can be reflected across the axes and the origin, affecting the signs of their coordinates and corresponding trigonometric values.

• Unit Circle: The circle of radius 1 centered at the origin.

• Reflections:

  • Across x-axis:

  • Across y-axis:

  • Across origin:

• Effect on Trigonometric Functions: Reflections change the sign of sine and/or cosine, depending on the axis.

• Example: The point reflected across the x-axis becomes .

Calculator Evaluation of Trigonometric Functions

Trigonometric function values for real numbers can be found using a scientific calculator, ensuring the correct mode (degrees or radians) is set.

• Steps:

  1. Set calculator to degree or radian mode as appropriate.

  2. Enter the value and select the desired trigonometric function.

  3. Round the answer as required.

• Example:

Graphing the Six Circular Functions and Their Properties

The six circular (trigonometric) functions—sine, cosine, tangent, cosecant, secant, and cotangent—have distinct graphs and properties.

• Key Properties:

  • Periodicity: Sine and cosine have period ; tangent and cotangent have period .

  • Amplitude: For sine and cosine, amplitude is the maximum value from the midline.

  • Asymptotes: Tangent, cotangent, secant, and cosecant have vertical asymptotes.

• Example Table:

Graph Transformations of Sine and Cosine Functions

Transformations include changes in amplitude, period, phase shift, and vertical shift. The general form is or .

• Amplitude (): is the height from the midline to the maximum or minimum.

• Period ():

• Phase Shift ():

• Vertical Shift (): Moves the graph up or down by units.

• Example: has amplitude 3, period , phase shift , and vertical shift 1.

Maximum and Minimum Values of Transformed Sine and Cosine Functions

The maximum and minimum values of a transformed sine or cosine function are determined by the amplitude and vertical shift.

• General Form:

• Maximum Value:

• Minimum Value:

• Example: For , maximum is , minimum is .

Applications Involving Transformed Sine and Cosine Functions

Transformed sine and cosine functions model periodic phenomena such as sound waves, tides, and seasonal temperatures.

• Steps:

  1. Identify amplitude, period, phase shift, and vertical shift from the context.

  2. Write the function in the form or .

  3. Use the model to answer questions about maximum, minimum, or specific values.

• Example: Modeling daylight hours over a year using a cosine function.