Section 4.2 The Unit Circle. Has a radius of 1 Center at the origin Defined by the equations: a) b)

Slides:



Advertisements
Similar presentations
Identify a unit circle and describe its relationship to real numbers
Advertisements

Chapter 4: Graphing & Inverse Functions
The Inverse Trigonometric Functions Section 4.2. Objectives Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
Section 5.3 Trigonometric Functions on the Unit Circle
7.4 Trigonometric Functions of General Angles
Sullivan Algebra and Trigonometry: Section 6.5 Unit Circle Approach; Properties of the Trig Functions Objectives of this Section Find the Exact Value of.
Chapter 4 Trigonometry Day 2 (Covers a variety of topics in ) 6 Notecards.
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.
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, 4.4 – The Unit Circle, Trig Functions The unit circle is defined by the equation x 2 + y 2 = 1. It has its center at the origin and radius 1. (0,
2.5 Properties of the Trig Functions
Section 5.3 Trigonometric Functions on the Unit Circle
Precalculus Section 7.5. Warmup Graph the function. State the Domain, Range, Asymptotes, and Period 1.f(x) = -2 tan(1/3 x) 2.f(x) = sec(2x) + 1.
January 19 th in your BOOK, 4.2 copyright2009merrydavidson.
12-2 Trigonometric Functions of Acute Angles
Bell Work Find all coterminal angles with 125° Find a positive and a negative coterminal angle with 315°. Give the reference angle for 212°.
Lesson 4.2. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (1,0) (0,1) (0,-1) (-1,0)
Lesson 4.2. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (1,0) (0,1) (0,-1) (-1,0)
Section 4.2 Trigonometric Functions: The Unit Circle
Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 4 Trigonometric Functions.
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.
Do Now: Graph the equation: X 2 + y 2 = 1 Draw and label the special right triangles What happens when the hypotenuse of each triangle equals 1?
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.
GRAPHS of Trig. Functions. We will primarily use the sin, cos, and tan function when graphing. However, the graphs of the other functions sec, csc, and.
14.2 The Circular Functions
Section 5.3 Evaluating 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.
1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved.
Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of.
Chapter 4 Trigonometric Functions The Unit Circle Objectives:  Evaluate trigonometric functions using the unit circle.  Use domain and period.
Trigonometric Functions Section 1.6. Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc.
Definitions of Trigonometric functions Let t be a real number and let (x,y) be the point on the unit circle corresponding to t Sin t = ycsc t = 1/y.
Aims: To know the relationship between the graphs and notation of cosine, sine and tan, with secant, cosecant and cotangent. To be able to state the domain.
Objectives : 1. To use identities to solve trigonometric equations Vocabulary : sine, cosine, tangent, cosecant, secant, cotangent, cofunction, trig identities.
4.2 Trig Functions of Acute Angles. Trig Functions Adjacent Opposite Hypotenuse A B C Sine (θ) = sin = Cosine (θ ) = cos = Tangent (θ) = tan = Cosecant.
Section 3 – Circular Functions Objective To find the values of the six trigonometric functions of an angle in standard position given a point on the terminal.
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.
Bellringer 3-28 What is the area of a circular sector with radius = 9 cm and a central angle of θ = 45°?
4.4 Day 1 Trigonometric Functions of Any Angle –Use the definitions of trigonometric functions of any angle –Use the signs of the trigonometric functions.
Right Triangle Trigonometry
Trigonometric Functions:Unit Circle
Lesson Objective: Evaluate trig functions.
The Other Trigonometric Functions
Section 4.2 The Unit Circle.
Introduction to the Six Trigonometric Functions & the Unit Circle
Trigonometric Functions: The Unit Circle Section 4.2
Trigonometric Functions: The Unit Circle 4.2
Pre-Calc: 4.2: Trig functions: The unit circle
Lesson 4.2 Trigonometric Functions: The Unit Circle
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
Right Triangle Ratios Chapter 6.
5.3 Properties of the Trigonometric Function
Graphs of Secant, Cosecant, and Cotangent
Trigonometric Functions: The Unit Circle
The Inverse Trigonometric Functions (Continued)
Introduction to College Algebra & Trigonometry
Trigonometric Functions: The Unit Circle
Trigonometric Functions: Unit Circle Approach
Trigonometric Functions: The Unit Circle 4.2
WArmup Rewrite 240° in radians..
Academy Algebra II THE UNIT CIRCLE.
Presentation transcript:

Section 4.2 The Unit Circle

Has a radius of 1 Center at the origin Defined by the equations: a) b)

The Unit Circle The real number t corresponds to the distance around the unit circle. Each real number t corresponds to a point (x, y) on the unit circle. Sin t = Cos t = Tan t = (1,0)(-1,0) (0,1) (0,-1) t t x y 1 t y x

The Unit Circle Determine the exact values of the six trig functions of the angle Ѳ Ѳ

The Unit Circle Determine the exact values of the six trig functions of the angle Ѳ Ѳ

The Unit Circle

Find the point (x, y) on the unit circle that corresponds to the real number t. t = y = Sin = x = Cos = The number t corresponds to the point (, ) º 60 º → Sin t = y, Cos t = x

The Unit Circle Find the point (x, y) on the unit circle that corresponds to the real number t. t = Sin t = Cos t = The real number t corresponds to the point

The Unit Circle Find the point (x, y) on the unit circle that corresponds to the real number t. Sin t = Cos t = Sin t = Cos t = t =

The Unit Circle

Section 4.2 The Unit Circle

Yesterday, we: a) Defined the unit circle b) Evaluated the exact value of the six trig functions of a point on the unit circle c) Found a point on the unit circle given the real number t d) Evaluated the sine, cosine, and tangent of the real number t Today: Evaluate all six trig functions of the real number t Use a functions period to evaluate the trig functions of t Use trig functions to evaluate other trig functions

The Unit Circle

Domain, Range, and Period Domain of the Sine function: All real numbers Domain of the Cosine function: All real numbers Range of these functions: [-1, 1] Period of both functions: 2π2π

Domain, Range, and Period What happens when you add 2 π to any value of t? Therefore: 1) Sin (t + 2 π ) = Sin t 2) Cos (t + 2 π ) = Cos t

Domain, Range, and Period Find the Sine Because = 2 π +, we have Sin = Sin (2 π + ) = Sin = ½

Domain, Range, and Period Find the Cosine Because = 2 π +, we have Cosin = Cos (2 π + ) = Cos = - ½

Practice CosSin = 2 π + → Cos = Cos → Cos = ½ Cos = ½ = 2 π + → Sin = Sin → Sin = Sin =

The Unit Circle

Even and Odd Functions Even FunctionsOdd Functions Cosine and Secant Cos (-t) = Cos t Sec (-t) = Sec t Sine and Cosecant Tangent and Cotangent Sin (-t) = - Sin t Csc (-t) = - Csc t Tan (-t) = - Tan t Cot (-t) = - Cot t

Even and Odd Functions Sin t =Cos t = - a) Sin (-t) = b) Csc (-t) = c) Sin ( π - t) = a) Sec (t) = b) Cos (t) = c) Cot (t + π ) = – – – –