Properties of the Trigonometric Functions

Domain and Range Remember: 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 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] 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  /2. The range is all real numbers. The domain of the tangent function is the set of all real numbers, except odd multiples of  /2. The range is all real numbers. The domain of the secant function is the set of all real numbers, except odd multiples of  /2. The range is (-∞, 1] u [1, ∞). The domain of the secant function is the set of all real numbers, except odd multiples of  /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 . The range is all real numbers. The domain of the cotangent function is the set of all real numbers except integral multiples of . The range is all real numbers. The domain of the cosecant function is the set of all real numbers except integral multiples of . The range is (-∞, 1] u [1, ∞) The domain of the cosecant function is the set of all real numbers except integral multiples of . The range is (-∞, 1] u [1, ∞)

Periodic Functions Definition: 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 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) f(θ + p) = f(θ)

Periodic Properties

Periodic Functions If sin θ = 0.3, find the value of sin θ + If sin θ = 0.3, find the value of sin θ + sin (θ + 2  ) + sin (θ + 4  ) sin (θ + 2  ) + sin (θ + 4  ) If tan θ = 3, find the value of tan θ + If tan θ = 3, find the value of tan θ + tan (θ +  ) + tan (θ + 2  ) tan (θ +  ) + tan (θ + 2  )

Signs of the Trigonometric Functions Table 5 p. 403 Table 5 p. 403 Remember the mnemonic (All – Quad I; Scientists – Quad II; Take – Quad III; Calculus – Quad IV) 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  and cos  < 0, name the quadrant in which the angle lies. If sin  and cos  < 0, name the quadrant in which the angle lies. If sin  < 0 and tan  < 0, name the quadrant in which the angle lies. If sin  < 0 and tan  < 0, name the quadrant in which the angle lies.

Fundamental Identities Reciprocal Identities: Reciprocal Identities: Quotient Identities:

Fundamental Identities Pythagorean 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. 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. 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 Definition Using Fundamental Identities Using Fundamental Identities

Even and Odd Properties

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