Properties of the Trigonometric Functions Domain and Range

Properties of the Trigonometric Functions

Domain and Range • Remember: •

Domain and Range • The domain of the sine function is all real numbers. The range of the sine function is [-1, 1] • The domain of the cosine function is all real numbers. The range of the cosine function is [-1, 1]

Domain and Range • The domain of the tangent function is the set of all real numbers, except odd multiples of p/2. The range is all real numbers. • The domain of the secant function is the set of all real numbers, except odd multiples of p/2. The range is (-∞, 1] u [1, ∞).

Domain and Range • The domain of the cotangent function is the set of all real numbers except integral multiples of p. The range is all real numbers. • The domain of the cosecant function is the set of all real numbers except integral multiples of p. The range is (-∞, 1] u [1, ∞)

Periodic Functions • Definition: • A function f is called periodic if there is a positive number p such that, whenever θ is in the domain of f, so is θ + p, and • f(θ + p) = f(θ)

Periodic Properties

Periodic Functions • If sin θ = 0. 3, find the value of sin θ + • sin (θ + 2 p) + sin (θ + 4 p) • If tan θ = 3, find the value of tan θ + • tan (θ + p) + tan (θ + 2 p)

Signs of the Trigonometric Functions • Table 5 p. 403 • Remember the mnemonic (All – Quad I; Scientists – Quad II; Take – Quad III; Calculus – Quad IV)

Finding the Quadrant in Which an Angle Lies • If sin q < 0 and cos q < 0, name the quadrant in which the angle lies. • If sin q < 0 and tan q < 0, name the quadrant in which the angle lies.

Fundamental Identities • Reciprocal Identities: Quotient Identities:

Fundamental Identities • Pythagorean Identities:

Finding Exact Values of A Trig Expression Find the other four trig functions using identities and/or unit circle

Find the Exact Value of Trig Functions • Find the exact value of each expression. Do not use a calculator.

Given One Value of a Trig Function, Find the Remaining Ones • Given that tan θ=½ and sin θ < 0, find the exact value of each of the remaining five trig functions of θ. • Using Definition • Using Fundamental Identities

Even and Odd Properties

Properties of Trig Functions • On-line Examples • On-line Tutorial