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MAC1114 Lecture Note Outlines_Eichler Lecture 5 (L5): Trig Functions of Any Angle and Unit Circle Textbook Section: 4.4 and 4.2 Review: Six trigonometric functions for an acute angle, 𝜽: Problem: Consider the obtuse angle, 5𝜋 6 , in standard position on the coordinate plane. Let (−2√3, 2) be a point on the terminal side of the angle. How can we define the six trigonometric functions for the angle 5𝜋 6 ? Reference Angles: If 𝜃 is an angle in standard position on the coordinate plane, then its _____________________________________________ is the acute angle formed by 𝜃′𝑠 terminal side and the ________________________________________. These angles can be used to find the values of the six trigonometric functions for an angle 𝜃 in standard position. 1 MAC1114 Lecture Note Outlines_Eichler Problem: Find the sine, cosine, and tangent values for the angles 5𝜋 4 and 5𝜋 3 . Problem: Consider an angle, 𝜃, in standard position that does not have a special angle as a reference angle. How can the six trigonometric functions be defined for any angle? Let 𝜃 be any angle in standard position and (𝑥 , 𝑦) be a point on its terminal side. Problem: How does the quadrant affect the sign of the trigonometric function values? 2 MAC1114 Lecture Note Outlines_Eichler Review: Equation of a Circle with center (h, k) and radius, r: Unit Circle: The ______________________________ is a circle centered at the _______________________ with a _____________________ of one. Diagram and Algebraic Representation: Recall how radians were defined as the intercepted angle of a circle with radius, r, and arc length, s, as ___________________________________. Imagine the real number line wrapping around the unit circle. Each number on the number line would correspond with a _______________________________ on the plane. Each number on the number line also corresponds to a central angle of the unit circle. Recall the formula for arc length, s, is _____________________________________. So each point along the circle is an ___________________________________ intercepted by a ______________________________ in radians. Diagram: 3 MAC1114 Lecture Note Outlines_Eichler Trigonometric Functions of Real Numbers: Since each real number along the unit circle is defined as ____________________________ with radius being ______________ , we can define the trigonometric functions for each real number along the unit circle. Remember each real number along the unit circle also represents a ______________________________ on the coordinate plane. Definitions for Trigonometric Functions: Let t represent a real number along the unit circle and (𝑥, 𝑦) is the point corresponding to the real number, t, along the unit circle. Then the trigonometric functions for the real number, t, are defined as follows: CAUTION!! TRIG FUNCTIONS ARE FUNCTIONS OF ANGLES!! ON THE UNIT CIRCLE BECAUSE THE RADIUS IS ONE THE ARC LENGTH EQUALS THE ANGLE MEASURE. THIS WILL NOT ALWAYS HAPPEN!!!!!!!!!!!! 4 MAC1114 Lecture Note Outlines_Eichler Problem: Consider the real number 𝑡 = 𝜋 6 on the unit circle. What coordinate, (𝑥, 𝑦), does t correspond to? Problem: What are the corresponding coordinates along the unit circle for the real numbers: 0 , 𝜋 2 ,𝜋 , 3𝜋 2 , 2𝜋 . Problem: Determine the exact values of the trigonometric functions of the real number t, 3 4 5 5 given that t corresponds to the point ( , − ). 5 MAC1114 Lecture Note Outlines_Eichler The unit circle can be divided into arcs corresponding to the real number t values of: Diagram: Domain and Range of Sine and Cosine Functions: We discussed earlier that sine and cosine could never be greater than __________________. Then a lower bound for sine and cosine is ___________________________. This means that the _____________________________ of sine and cosine is _____________________________________. Now we consider that the unit circle is made by wrapping the real line around the circle. The sine and cosine functions can continue take values as we continue to move around the circle. That means that the _____________________________ of the sine and cosine functions is ________________________________________. 6 MAC1114 Lecture Note Outlines_Eichler Period of the Sine and Cosine Functions: A function is said to be _________________________________ if there is a positive real number, a, such that ______________________________________________ for all x in the domain of the function. The smallest number, a, such that the function is periodic is called the _______________________________. Problem: What is the period of the sine and cosine functions? Review: Odd function: Even function: Problem: Are the six trigonometric functions odd or even? 7