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Algebra and Trig. I 4.2 – Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is . We can use the following formula for the length of a circular arc, , to find the length of the intercepted arc. where 1 is the radius of the circle and t is the radian measure of the central angle. Thus the length of the intercepted arc is θ, which is also the radian measure of the central angle. *So in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. P(x,y) (1,0) 0 0 θ (1,0) θ When θ is positive the point P is reached by moving counterclockwise along the unit circle from (1,0) When θ is negative the point P is reached by moving clockwise along the unit circle from (1,0) P(x,y) 1|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College 2|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College The Six Trigonometric Functions – The inputs of these six functions are θ and the outputs involve the point on the unit circle corresponding to t and the coordinates of this point. Trig. Functions have names that are words, rather than single letters such as . For example, the sine of θ is the ycoordinate of the point P on the unit circle. The real number y depends on the letter θ and thus is a function of t. really means , where sine is the name of the function and θ, a real number, is an input. Name sine cosine tangent Abbreviation sin cos tan Name cosecant secant cotangent Abbreviation csc sec cot Definitions of the Trigonometric Functions in Terms of Any Circle: If θ is a real number and is a point on the unit circle that corresponds to θ, and r is the radius then 3|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Definitions of the Trigonometric Functions in Terms of a Unit Circle: If θ is a real number and is a point on the unit circle that corresponds to θ, where r is the radius which is 1 then Finding Values of the Trigonometric Functions P( , ) θ θ 0 If θ is a real number equal to the length of the intercepted arc of an angle that measures θ radians and (1,0) is the point on the unit circle that corresponds to θ. Find the values of the six trig. functions at t. Don’t forget to rationalize if needed. a) c) b) d) 4|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College f) e) Example – P= Use the figure to the left to determine the values of the six trig. functions at . 0 (1,0) a) b) g) h) i) j) 5|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Domain and Range of Sine and Cosine Functions – Example – Find Example – Find 6|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Trigonometric Functions at Even and Odd Trigonometric Functions We know that a function is even if and odd if . We can show that the cosine function is even and sine is odd. By definition, the coordinates of the points P and Q are as follows: P (1, 0) 0 Q In the above figure the x-coordinates of P and Q are the same, thus thus the cosine function is even. In the above figure the y-coordinates of P and Q are negatives of each other, thus thus the sine function is odd 7|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Even and Odd Trigonometric Functions Only cosine and secant functions are even all other are odd. Example – Find the value of each trigonometric function: a) b) Fundamental Identities Trigonometric identities are equations that are true for all real numbers for which the trigonometric expressions in the equations are defined. Reciprocal Identities 8|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Quotient Identities Example – Given and , find the value of each of the four remaining trigonometric functions. 9|Page Hannah Province – Mathematics Department – Southwest Tennessee Community College Pythagorean Identities – Example – Given and using a trigonometric identity. , find the value of Example – Given and using a trigonometric identity. , find the value of 10 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College Periodic Functions A function f is periodic if there exists a positive number p such that for all in the domain of f. The smallest positive number p for which f is periodic is called the period of f. Periodic Properties of the Sine and Cosine Functions The sine and cosine functions are periodic functions and have period 2π. The secant and cosecant functions have period 2π. Periodic Properties of the Tangent and Cotangent Functions The tangent and cotangent functions are periodic functions and have period π. Example – Find the value of each trigonometric function. a) c) b) d) 11 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College Repetitive Behavior of the Sine, Cosine and Tangent Functions For any integer n and real number , Evaluating Trig. Functions with a Calculator a) b) 12 | P a g e Hannah Province – Mathematics Department – Southwest Tennessee Community College