Trigonometric Functions

The three primary trigonometric functions are sine, cosine, and tangent. These functions are defined based on the ratios of the sides of a right-angled triangle.

Sine Function

The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

sin(θ) = Opposite side / Hypotenuse

Cosine Function

The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

cos(θ) = Adjacent side / Hypotenuse

Tangent Function

The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

tan(θ) = Opposite side / Adjacent side

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables within their domains. Some common trigonometric identities include Pythagorean and Reciprocal identities.

Pythagorean Identities

• sin^2(θ) + cos^2(θ) = 1
• 1 + tan^2(θ) = sec^2(θ)
• 1 + cot^2(θ) = csc^2(θ)

Reciprocal Identities

• csc(θ) = 1/sin(θ)
• sec(θ) = 1/cos(θ)
• cot(θ) = 1/tan(θ)

Trigonometric Functions and Graphs

The trigonometric functions can also be represented graphically. The sine and cosine functions, for example, produce periodic wave-like graphs. The amplitude, period, and phase shift of these functions can be analyzed to understand their behavior.

Study Guide

• Understand the definitions of sine, cosine, and tangent functions.
• Practice using the trigonometric functions to calculate angles and side lengths in right-angled triangles.
• Memorize the Pythagorean and reciprocal identities.
• Learn to sketch and interpret graphs of sine and cosine functions.