Basics of Trigonometry

Basics of Trigonometry

• Definition: Study of relationships between angles and sides of triangles, particularly right triangles.
• Primary Functions: Sine (sin), Cosine (cos), Tangent (tan).

Right Triangle Ratios

• Sine (sin θ): Opposite side / Hypotenuse
• Cosine (cos θ): Adjacent side / Hypotenuse
• Tangent (tan θ): Opposite side / Adjacent side
  • Relationship: tan θ = sin θ / cos θ

Reciprocal Functions

• Cosecant (csc θ): 1 / sin θ
• Secant (sec θ): 1 / cos θ
• Cotangent (cot θ): 1 / tan θ = cos θ / sin θ

Pythagorean Identity

• For any angle θ:
  • sin² θ + cos² θ = 1
• Pythagorean Theorem: In a right triangle, a² + b² = c² (a and b are legs, c is hypotenuse).

Unit Circle

• Definition: A circle with radius 1 centered at the origin (0,0) on a coordinate plane.
• Coordinates: (cos θ, sin θ) give the position of a point on the unit circle.
• Important Angles:
  • 0° (0 radians): (1, 0)
  • 90° (π/2 radians): (0, 1)
  • 180° (π radians): (-1, 0)
  • 270° (3π/2 radians): (0, -1)
  • 360° (2π radians): (1, 0)

Trigonometric Functions Graphs

• Sine Wave: Oscillates between -1 and 1.
• Cosine Wave: Similar shape, shifted left by 90°.
• Tangent Function: Periodic with vertical asymptotes; unbounded.

Angle Measures

• Degrees and Radians:
  • Full circle: 360° = 2π radians
  • Conversion: radians = degrees × (π/180)

Law of Sines

• Formula: a/sin A = b/sin B = c/sin C
  • Relates the lengths of sides of a triangle to the sines of its angles.

Law of Cosines

• Formula: c² = a² + b² - 2ab * cos C
  • Used for finding a side of a triangle when angles and one side are known.

Applications

• Real-world: Navigation, architecture, physics, engineering.
• Modeling: Waves, oscillations, and circular motions.

Key Identities

• Angle Sum/Difference:
  • sin(A ± B) = sin A cos B ± cos A sin B
  • cos(A ± B) = cos A cos B ∓ sin A sin B
  • tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Special Angles

• Angles 30°, 45°, 60° have known sine, cosine, and tangent values:
  • 30°: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
  • 45°: sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
  • 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Summary

• Trigonometry is essential in mathematics, with broad applications in various fields.
• Understanding foundational concepts and functions is crucial for further study and application.

Basics of Trigonometry

• Trigonometry studies the relationships between angles and sides of triangles, especially right triangles.
• The core functions are sine (sin), cosine (cos), and tangent (tan).

Right Triangle Ratios

• Sine (sin θ) is the ratio of the opposite side to the hypotenuse of a right triangle.
• Cosine (cos θ) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan θ) is the ratio of the opposite side to the adjacent side.
• tan θ can also be calculated as sin θ / cos θ.

Reciprocal Functions

• Cosecant (csc θ) is the reciprocal of sine: 1 / sin θ.
• Secant (sec θ) is the reciprocal of cosine: 1 / cos θ.
• Cotangent (cot θ) is the reciprocal of tangent: 1 / tan θ, which is equivalent to cos θ / sin θ.

Pythagorean Identity

• For any angle θ, the equation sin² θ + cos² θ = 1 holds true.
• This is closely related to the Pythagorean Theorem in right triangles: a² + b² = c², where a and b are the legs and c is the hypotenuse.

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane.
• The coordinates of a point on the unit circle are (cos θ, sin θ), where θ is the angle formed from the positive x-axis to the point.
• Key angles on the unit circle include:
  • 0° (0 radians): (1, 0)
  • 90° (π/2 radians): (0, 1)
  • 180° (π radians): (-1, 0)
  • 270° (3π/2 radians): (0, -1)
  • 360° (2π radians): (1, 0)

Trigonometric Functions Graphs

• The sine wave graph oscillates between -1 and 1, with a period of 2π.
• The cosine wave is similar in shape to the sine wave but is shifted to the left by 90°.
• The tangent function has a periodic graph with vertical asymptotes at odd multiples of π/2 and is unbounded.

Angle Measures

• Angles can be measured in degrees or radians.
• A full circle is 360° or 2π radians.
• To convert from degrees to radians, multiply by π/180.

Law of Sines

• The Law of Sines states that the ratio of a side of a triangle to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C.

Law of Cosines

• The Law of Cosines is used to find a side of a triangle when two sides and the angle between them are known.
• The formula is c² = a² + b² - 2ab * cos C, where c is the side opposite angle C.

Applications

• Trigonometry has many practical applications in various fields like navigation, architecture, physics, and engineering.
• It is useful for modeling wave patterns, oscillations, and circular motion.

Key Identities

• Angle Sum/Difference Identities:
  • sin(A ± B) = sin A cos B ± cos A sin B
  • cos(A ± B) = cos A cos B ∓ sin A sin B
  • tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Special Angles

• Some common angles have known sine, cosine, and tangent values.
  • 30°: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
  • 45°: sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
  • 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Summary

• Trigonometry is a fundamental concept in mathematics with numerous applications in real-world scenarios.
• Understanding the basic functions and principles is crucial for further exploration and practical use of trigonometry.

Explore the fundamental concepts of trigonometry, focusing on the relationships between angles and sides of triangles, particularly right triangles. This quiz covers primary functions, triangle ratios, reciprocal functions, identities, and the unit circle. Test your understanding of these essential mathematical principles.