Trigonometry: Angles, Triangles, and Ratios

Trigonometry

Angles and Triangles

• Angle measurements: degrees, radians, and gradients
• Types of angles: acute, right, obtuse, straight, and reflex angles
• Triangle properties: equilateral, isosceles, scalene, and right-angled triangles

Trigonometric Ratios

• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): opposite side / adjacent side
• Cotangent (cot): adjacent side / opposite side
• Secant (sec): hypotenuse / opposite side
• Cosecant (csc): hypotenuse / adjacent side

Trigonometric Identities

• Pythagorean identity: sin^2(A) + cos^2(A) = 1
• Sum and difference formulas: sin(A+B) = sin(A)cos(B) + cos(A)sin(B), etc.
• Double angle formulas: sin(2A) = 2sin(A)cos(A), etc.

Graphs of Trigonometric Functions

• Sine and cosine graphs: periodic, symmetric, and amplitude
• Tangent graph: periodic, but not symmetric, and vertical asymptotes

Solving Triangles

• Right-angled triangles: use trigonometric ratios to find unknown sides or angles
• Oblique triangles: use law of sines or law of cosines to find unknown sides or angles

Applications of Trigonometry

• Real-world problems: height and distance, navigation, physics, and engineering
• Modeling periodic phenomena: sound waves, light waves, and electrical signals

Angle Measurements

• Angle measurements can be expressed in degrees, radians, or gradients.
• Degrees are the most common unit of measurement, with 360 degrees in a full circle.

Types of Angles

• Acute angles are less than 90 degrees.
• Right angles are exactly 90 degrees.
• Obtuse angles are greater than 90 degrees but less than 180 degrees.
• Straight angles are exactly 180 degrees.
• Reflex angles are greater than 180 degrees but less than 360 degrees.

Triangle Properties

• Equilateral triangles have all sides of equal length.
• Isosceles triangles have two sides of equal length.
• Scalene triangles have all sides of different lengths.
• Right-angled triangles have one right angle (90 degrees).

Trigonometric Ratios

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• Cotangent (cot) is the ratio of the adjacent side to the opposite side.
• Secant (sec) is the ratio of the hypotenuse to the opposite side.
• Cosecant (csc) is the ratio of the hypotenuse to the adjacent side.

Trigonometric Identities

• The Pythagorean identity is sin^2(A) + cos^2(A) = 1.
• Sum and difference formulas are used to find trigonometric values of angles.
• Double angle formulas are used to find trigonometric values of double angles.

Graphs of Trigonometric Functions

• Sine and cosine graphs are periodic, symmetric, and have amplitude.
• Tangent graphs are periodic, but not symmetric, and have vertical asymptotes.

Solving Triangles

• Right-angled triangles can be solved using trigonometric ratios.
• Oblique triangles can be solved using the law of sines or law of cosines.

Applications of Trigonometry

• Trigonometry is used to solve real-world problems involving height and distance, navigation, physics, and engineering.
• Trigonometry is used to model periodic phenomena such as sound waves, light waves, and electrical signals.