Trigonometric Functions: The Unit Circle 4.2

Slides:



Advertisements
Similar presentations
The Inverse Trigonometric Functions Section 4.2. Objectives Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
Advertisements

Trigonometric Functions: The Unit Circle 1.2. Objectives  Students will be able to identify a unit circle and describe its relationship to real numbers.
Sullivan Algebra and Trigonometry: Section 6.5 Unit Circle Approach; Properties of the Trig Functions Objectives of this Section Find the Exact Value of.
7.3 Trigonometric Functions of Angles. Angle in Standard Position Distance r from ( x, y ) to origin always (+) r ( x, y ) x y  y x.
Properties of the Trigonometric Functions. Domain and Range Remember: Remember:
Pre calculus Problem of the Day Homework: p odds, odds, odds On the unit circle name all indicated angles by their first positive.
5.2-The Unit Circle & Trigonometry. 1 The Unit Circle 45 o 225 o 135 o 315 o 30 o 150 o 110 o 330 o π6π6 11π 6 5π65π6 7π67π6 7π47π4 π4π4 5π45π4 3π43π4.
4.2 Trigonometric Function: The Unit circle. The Unit Circle A circle with radius of 1 Equation x 2 + y 2 = 1.
Trigonometric Functions Of Real Numbers
Copyright © Cengage Learning. All rights reserved. 4 Trigonometric Functions.
Section 4.2 Trigonometric Functions: The Unit Circle
Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions.
13.7 (part 2) answers 34) y = cos (x – 1.5) 35) y = cos (x + 3/(2π)) 36) y = sin x –3π 37) 38) y = sin (x – 2) –4 39) y = cos (x +3) + π 40) y = sin (x.
WEEK 10 TRIGONOMETRIC FUNCTIONS TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS; PERIODIC FUNCTIONS.
5.3 Properties of the Trigonometric Function. (0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)
4.2 Trigonometric Functions (part 2) III. Trigonometric Functions. A) Basic trig functions: sine, cosine, tangent. B) Trig functions on the unit circle:
Chapter 4 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Trigonometric Functions: The Unit Circle.
Graphs of the Trig Functions Objective To use the graphs of the trigonometric functions.
Chapter 5 – Trigonometric Functions: Unit Circle Approach Trigonometric Function of Real Numbers.
5.3 The Unit Circle. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be So points on this circle.
5.2 – Day 1 Trigonometric Functions Of Real Numbers.
Chapter 4 Trigonometric Functions The Unit Circle Objectives:  Evaluate trigonometric functions using the unit circle.  Use domain and period.
Warm up Solve for the missing side length. Essential Question: How to right triangles relate to the unit circle? How can I use special triangles to find.
Copyright © Cengage Learning. All rights reserved. 4.2 Trigonometric Functions: The Unit Circle.
Sullivan Precalculus: Section 5.3 Properties of the Trig Functions Objectives of this Section Determine the Domain and Range of the Trigonometric Functions.
Trigonometry Section 4.2 Trigonometric Functions: The Unit Circle.
SFM Productions Presents: Another saga in your continuing Pre-Calculus experience! 4.6 Graphs of other Trigonometric Functions.
Bellringer 3-28 What is the area of a circular sector with radius = 9 cm and a central angle of θ = 45°?
Section 4.2 The Unit Circle. Has a radius of 1 Center at the origin Defined by the equations: a) b)
Math IV Warm Up Draw a circle on your paper. Fill in the degrees of the entire unit circle.
Chapter 5 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Trigonometric Functions of Real Numbers; Periodic Functions.
The Unit Circle with Radian Measures. 4.2 Trigonometric Function: The Unit circle.
Right Triangle Trigonometry
Trigonometric Functions:Unit Circle
Lesson Objective: Evaluate trig functions.
Section 4.2 The Unit Circle.
Introduction to the Six Trigonometric Functions & the Unit Circle
Trigonometric Functions: The Unit Circle Section 4.2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Trigonometric Functions: The Unit Circle 4.2
Pre-Calc: 4.2: Trig functions: The unit circle
Chapter 1 Angles and The Trigonometric Functions
Evaluating Angles.
Lesson 4.2 Trigonometric Functions: The Unit Circle
The Unit Circle The two historical perspectives of trigonometry incorporate different methods of introducing the trigonometric functions. Our first introduction.
2. The Unit circle.
LESSON ____ SECTION 4.2 The Unit Circle.
Copyright © Cengage Learning. All rights reserved.
THE UNIT CIRCLE SECTION 4.2.
Trigonometric Functions: The Unit Circle (Section 4-2)
Warm-Up: February 3/4, 2016 Consider θ =60˚ Convert θ into radians
Warm-Up: Give the exact values of the following
Chapter 8: The Unit Circle and the Functions of Trigonometry
5.2 Trigonometric Functions of Real Numbers
5.3 Properties of the Trigonometric Function
Graphs of Secant, Cosecant, and Cotangent
Trigonometric Functions: The Unit Circle
2) Find one positive and one negative coterminal angle to
Graph of Secant, Cosecant, and Cotangent
The Inverse Trigonometric Functions (Continued)
Introduction to College Algebra & Trigonometry
Trigonometric Functions
Trigonometric Functions: The Unit Circle
5.2 Trigonometric Functions: Unit Circle Approach
Trigonometric Functions: Unit Circle Approach
Evaluating Angles.
Precalculus Essentials
5.2 Trigonometric Functions: Unit Circle Approach
Academy Algebra II THE UNIT CIRCLE.
Copyright © Cengage Learning. All rights reserved.
Presentation transcript:

Trigonometric Functions: The Unit Circle 4.2

Unit Circle The unit circle is a circle of radius 1 with its center at the origin.

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

Points on the Unit Circle

Example 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

The Domain and Range of the Sine and Cosine Functions 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.

Evaluating Trigonometric Functions Evaluate the six trig functions at each real number. (a) t=л/6 (b) t=5л/4 (c) t=0 (d) t=л

Evaluate the 6 Trig Functions at t=-л/3

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

Example If sin t = 2/5 and cos t = 21/5, find the remaining four trig functions

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. sin  = sin 3

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 . tan  = tan 2