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.

Test your understanding of trigonometry concepts including angle measurements, types of angles, triangle properties, and trigonometric ratios such as sine, cosine, and tangent.