Basic Trigonometric Ratios and Identities

Basic Trigonometric Ratios

• Sine (sin): The ratio of the side opposite to an acute angle to the hypotenuse in a right-angled triangle.
• Cosine (cos): The ratio of the side adjacent to an acute angle to the hypotenuse in a right-angled triangle.
• Tangent (tan): The ratio of the side opposite to an acute angle to the side adjacent to that angle in a right-angled triangle.

Relationship Between Trigonometric Ratios

• Identities: These are equations that are true for all values of the angle. Key identities include:
• sin²(θ) + cos²(θ) = 1
• tan²(θ) = sin²(θ) / cos²(θ)

Trigonometric Functions

• Periodicity: Functions like sine, cosine, and tangent repeat their values at regular intervals.
• Domain and Range: The input values (domain) and output values (range) for each trigonometric function.
• Graphs: Visual representations of trigonometric functions, showing the periodicity and patterns.

Trigonometric Identities

• Useful for simplifying expressions and solving equations.
• Pythagorean Identities (as mentioned above) are fundamental to simplifying and solving trigonometric problems.
• Other identities include double-angle identities, half-angle identities, sum and difference identities. These relate trigonometric functions of multiple angles to functions of single angles.

Angles and Units of Measurement

• Degrees and Radians: Different units used to measure angles.
• Degrees: 360 degrees in a full circle.
• Radians: 2π radians in a full circle. The relationship between degrees and radians is 180° = π radians.

Applying Trigonometry

• Solving Right Triangles: Using trigonometric ratios to find unknown sides and angles in right-angled triangles.
• Real-World Applications: Trigonometry appears in numerous fields such as:
• Surveying to determine distances and heights.
• Navigation (especially celestial navigation).
• Engineering (architecture, construction).
• Physics (wave motion, vectors).

Inverse Trigonometric Functions

• Reversal of trigonometric functions. They determine an angle when you know a trigonometric ratio.
• Notation: sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x) are used for the inverse functions.

Special Angles

• Important values such as sin(30°), cos(45°), tan(60°), and their corresponding values in radians. Knowledge of these special angles is critical for quick calculation.

Graphs of Trigonometric Functions

• Understanding the shape of the graphs is crucial for visualization and applying trigonometric concepts.
• Key features of the graphs include amplitude, period, and phase shift.

Trigonometric Equations

• Solving equations involving trigonometric functions often requires using identities, properties of the functions themselves, or a combination of these.
• Identifying the general solutions: recognizing that multiple angles that satisfy the equation exist.

Trigonometric Ratios: Beyond Right-Angled Triangles

• Extending the concepts to encompass any triangle via the Law of Sines and Cosines
• Law of Sines: This law relates the sides of a triangle to the sines of the corresponding angles.
• Law of Cosines: This law relates the sides of a triangle to the cosine of one of its angles. This law allows the solving of any type of triangle.

Further Applications

• Vectors in 2D and 3D space rely extensively on trigonometric ratios.
• Complex Numbers have trigonometric representations.

This quiz covers the foundational concepts of basic trigonometric ratios such as sine, cosine, and tangent. Additionally, it explores the relationships between these ratios, periodicity of trigonometric functions, and key identities. Engage with the material to enhance your understanding of trigonometry.