Basic Trigonometric Functions

Basic Trigonometric Functions

• Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles.
• It's primarily used to calculate unknown angles and sides in right-angled triangles.
• The fundamental trigonometric functions relate an angle in a right-angled triangle to the ratios of the sides.
• These functions are sine (sin), cosine (cos), and tangent (tan).
• Other related functions are cosecant (csc), secant (sec), and cotangent (cot).

Sine, Cosine, and Tangent

• Sine (sin θ): The ratio of the side opposite to the angle to the hypotenuse.

• Cosine (cos θ): The ratio of the side adjacent to the angle to the hypotenuse.

• Tangent (tan θ): The ratio of the side opposite to the angle to the side adjacent to the angle.

• These functions are related by reciprocal relationships:

  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ

Trigonometric Identities

• Trigonometric identities are equations that are true for all values of the angles for which the functions are defined.

• Basic identities include:

  • sin² θ + cos² θ = 1
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ

• Other important identities relate functions of complementary and supplementary angles.

Applications of Trigonometry

• Trigonometry is used in various fields, including:
  • Navigation: Calculating distances and directions.
  • Engineering: Designing structures and determining heights.
  • Astronomy: Calculating distances to stars and planets.
  • Surveying: Determining land area and property boundaries.
  • Physics: Modeling motion and waves.
  • Computer graphics: Representing 3D objects.

Angles and Their Measurements

• Angles can be measured in degrees or radians.
• A full circle is 360 degrees (or 2π radians).
• A right angle is 90 degrees (or π/2 radians).
• Converting between degrees and radians is important for calculations.

Special Triangles

• 30-60-90 Triangles: Special ratios for the sides corresponding to the angles.
• 45-45-90 Triangles: Another set of special ratios for sides.
• Memorizing the ratios of these triangle types simplifies calculations and saves time in solving problems.

Unit Circle

• The unit circle is a circle with a radius of 1.
• Useful for visualizing trigonometric functions on a coordinate plane.
• Points on the unit circle correspond to the sine and cosine values of angles.

Trigonometric Functions of General Angles

• Trigonometric functions can be extended to any angle (positive or negative), by utilizing the unit circle concept.
• Reference angles are used to determine the sign and the magnitude of the value for functions.
• Quadrant rules for determining positive/negative signs of sin, cos and tan for different angle ranges should be understood.

Graphs of Trigonometric Functions

• The graphs of sine, cosine, and tangent repeat over intervals of 2π (or 360 degrees), displaying periodic properties.

Inverse Trigonometric Functions

• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle corresponding to a given trigonometric value such as arcsin(1/2) = 30° or π/6 radians.

Solving Trigonometric Equations

• Methods for solving equations involving trigonometric functions.
  • Using trigonometric identities
  • Factoring
  • Using the unit circle or special triangles.
• Applying these skills to find angle values that satisfy given equations.

This quiz covers the fundamental concepts of basic trigonometric functions including sine, cosine, and tangent. Learn how to apply these functions to calculate sides and angles in right-angled triangles. It also includes understanding the reciprocal relationships and identities that relate to these functions.