Trigonometric Functions: The Unit Circle 1.2

Objectives  Students will be able to identify a unit circle and describe its relationship to real numbers.  Students will be able to use a unit circle to evaluate trigonometric functions.  Students will be able to use the domain and period to evaluate sine and cosine functions.  Students will be able to use a calculator to evaluate trigonometric functions.

The Unit Circle (1, 0) (0, 1) (0, -1) (-1, 0)

Definitions of the Trigonometric Functions in Terms of a Unit Circle If t is a real number and (x, y) is a point on the unit circle that corresponds to t, then

Use the Figure to find the values of the trigonometric functions at t=  /2. Solution: The point P on the unit circle that Corresponds to t=  /2 has coordinates (0,1). We use x=0 and y=1 to find the Values of the trigonometric functions

T HE D OMAIN AND R ANGE OF THE S INE AND C OSINE F UNCTIONS A ND T HEIR P ERIOD The domain of the sine function and the cosine function is the set of all real numbers The range of these functions is the set of all real numbers from -1 to 1, inclusive. The period is 2π. This means it repeats every Periodic: f(t+c)=f(t) where c= 2π. Page 152 # 36, 42

Definition of a Periodic Function A function f is periodic if there exists a positive number p such that f(t + p) = f(t) For all t in the domain of f. The smallest number p for which f is periodic is called the period of f.

Periodic Properties of the Sine and Cosine Functions sin(t + 2  ) = sin t and cos(t + 2  ) = cos t The sine and cosine functions are periodic functions and have period 2 .

Periodic Properties of the Tangent and Cotangent Functions tan(t +  ) = tan t and cot(t +  ) = cot t The tangent and cotangent functions are periodic functions and have period .

Even and Odd Trigonometric Functions The cosine and secant functions are even. cos(-t) = cos tsec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(-t) = -sin tcsc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t

E XAMPLE Use the value of the trigonometric function at t =  /4 to find sin (-  /4 ) and cos(-  /4 ). Solution: sin(-t) = -sin t, so sin(-  /4 ) = -sin(  /4 ) = -  2/2 cos(-t) = cos t, so cos(-  /4 ) = cos(  /4 ) =  2/2 Try it: Pg. 151 # 38, 40, 46, 48, 50, 56 Homework: # 5 – 57 odd

Right Angle Trigonometry

Objectives  Students will be able to evaluate trigonometric functions of acute angles.  Students will be able to use fundamental trigonometric identities.  Students will be able to use a calculator to evaluate trigonometric functions.  Students will be able to use trigonometric functions to model and solve real life problems.

The Six Trigonometric Functions Hypotenuse Side opposite . Side adjacent to . The figure below shows a right triangle with one of its acute angles labeled . The side opposite the right angle is known as the hypotenuse. The other sides of triangle are described by the position relative to the acute angle . One side is opposite  and one is adjacent to . 

Hyp Opp Adj  R IGHT T RIANGLE D EFINITIONS OF T RIGONOMETRIC F UNCTIONS How does compare to the unit circle? Page 160 #8, 12

45° 1 1 √2/2 30° 60° √3

Reciprocal IdentitiesReciprocal Identities

Quotient Identities

Examples Page 161 #32, 38, 44, 46, 58, 62, 66, 70 Homework: 5 – 47 odd, 57 – 67 odd