Trigonometric Functions and Their Properties

This page delves into the main trigonometric functions and their characteristics.

The primary trigonometric functions are introduced:

• Sine (sin) • Cosine (cos) • Tangent (tan) • Cotangent (cot)

Definition: • sin θ = opposite side / hypotenuse • cos θ = adjacent side / hypotenuse • tan θ = opposite side / adjacent side = sin θ / cos θ • cot θ = adjacent side / opposite side = cos θ / sin θ

Key properties of these functions are discussed:

• Domain and range of each function • Periodicity • Relationship between functions (e.g., tan θ = sin θ / cos θ)

Highlight: The fundamental trigonometric identity: sin²θ + cos²θ = 1

The section also covers the behavior of these functions in different quadrants of the coordinate plane, emphasizing sign changes and special angle values.

Example: At 45°, sin θ = cos θ = 1/√2, and tan θ = 1

Trigonometric Ratios and Special Angles

This section focuses on trigonometric ratios for common angles and methods for solving right-angled triangles.

A table of trigonometric ratios for special angles (30°, 45°, 60°) is provided:

Highlight: • For 30°: sin = 1/2, cos = √3/2, tan = 1/√3 • For 45°: sin = cos = 1/√2, tan = 1 • For 60°: sin = √3/2, cos = 1/2, tan = √3

The concept of complementary angles is introduced, explaining how sine and cosine of complementary angles are related.

Definition: Complementary angles are two angles that add up to 90°. For complementary angles α and β, sin α = cos β and cos α = sin β.

The section also covers the application of these ratios in solving right-angled triangles and real-world problems.

Example: In a right-angled triangle with an angle of 30°, if the hypotenuse is 10 units, the opposite side is 5 units and the adjacent side is 5√3 units.

Trigonometric Identities and Formulas

This page covers important trigonometric identities and formulas essential for solving complex problems.

Key identities discussed include:

• Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ • Double angle formulas: sin 2θ = 2sin θ cos θ, cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ • Sum and difference formulas for sine, cosine, and tangent

Highlight: The sum formula for sine: sin$A + B$ = sin A cos B + cos A sin B

The section also introduces the concept of trigonometric equations and methods to solve them.

Example: Solving the equation sin x = sin 30° involves finding all angles that satisfy this condition, not just x = 30°.

Inverse Trigonometric Functions and Applications

The final section introduces inverse trigonometric functions and their applications.

Inverse trigonometric functions (arcsin, arccos, arctan, arccot) are defined and their properties discussed.

Definition: The inverse sine function, denoted as arcsin or sin⁻¹, is the inverse of the sine function restricted to the domain $-1, 1$ and range $-π/2, π/2$.

The section explains why the domain and range of trigonometric functions need to be restricted to define their inverses.

Highlight: The composition of a trigonometric function and its inverse yields the identity function: sin(arcsin x) = x for x in $-1, 1$.

Applications of inverse trigonometric functions in solving equations and real-world problems are presented.

Example: To solve the equation sin x = 0.5, we can use the inverse sine function: x = arcsin 0.5 = 30° (or π/6 radians).

The guide concludes with a brief overview of how these concepts are applied in more advanced topics in trigonometry and calculus.

Introduction to Trigonometry and the Unit Circle

This section introduces fundamental concepts in trigonometry, focusing on angle measurements and the unit circle.

The unit circle is defined as a circle with a radius of 1 unit centered at the origin (0,0) on the coordinate plane. It plays a crucial role in understanding trigonometric functions and their relationships.

Definition: A unit circle is a circle with a radius of 1 unit and its center at the origin (0,0) of the coordinate plane.

Angle measurements are discussed in both degrees and radians:

• 360° in a full circle corresponds to 2π radians • 1 radian is approximately 57.3°

Highlight: The relationship between degrees and radians is given by the formula: π radians = 180°

The coordinate axes on the unit circle are labeled: • x-axis is called the cosine axis • y-axis is called the sine axis

Example: On the unit circle, the point (cos θ, sin θ) represents the angle θ.