Copyright © Cengage Learning. All rights reserved. 6 Trigonometry Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 6.4 GRAPHS OF SINE AND COSINE FUNCTIONS Copyright © Cengage Learning. All rights reserved.

What You Should Learn Sketch the graphs of basic sine and cosine functions. Use amplitude and period to help sketch the graphs of sine and cosine functions. Sketch translations of the graphs of sine and cosine functions.

Basic Sine and Cosine Curves

Basic Sine and Cosine Curves    

Basic Sine and Cosine Curves five key points to sketch the graphs: the intercepts, maximum points, and minimum Figure 6.49

Example 1 – Using Key Points to Sketch a Sine Curve Sketch the graph of y = 2 sin x on the interval [–, 4]. Solution: Intercept Maximum Intercept Minimum Intercept

Exercise cont’d Exercise 39

Amplitude and Period

Amplitude = value from x-axis to maximum/minimum Amplitude and Period Amplitude = value from x-axis to maximum/minimum

Example 2: Scaling: Vertical Shrinking and Stretching Example 2 Exercise 41

Amplitude and Period period: value to complete 1 cycle, from 1st intercept to3rd intercept.

Example 3 – Scaling: Horizontal Stretching Sketch the graph of . Solution: The amplitude is 1. Moreover, because b = , the period is Substitute for b.

Example 3 – Solution cont’d divide the period-interval [0, 4] into four equal parts with the values , 2, and 3 to obtain the key points on the graph. Intercept Maximum Intercept Minimum Intercept (0, 0), (, 1), (2, 0), (3, –1), and (4, 0) The graph is shown in Figure 6.53. Figure 6.53

Translations of Sine and Cosine Curves

Translations of Sine and Cosine Curves

Example 5 – Horizontal Translation Sketch the graph of y = –3 cos(2 x + 4). Solution: The amplitude is 3 and the period is 2 / 2 = 1. By solving the equations 2 x + 4 = 0 2 x = –4 x = –2 and 2 x + 4 = 2

Example 5 – Solution 2 x = –2 x = –1 cont’d 2 x = –2 x = –1 you see that the interval [–2, –1] corresponds to one cycle of the graph. Dividing this interval into four equal parts produces the key points Minimum Intercept Maximum Intercept Minimum and

Example 5 – Solution The graph is shown in Figure 6.55. cont’d

Translations of Sine and Cosine Curves vertical translation for : y = d + a sin(bx – c) and y = d + a cos(bx – c). Shift d units upward for d > 0 Shift d units downward for d < 0 EXP 6

Graph of the Tangent Function

Graph of the Tangent Function Figure 6.59

Introduction We will learn how to use the fundamental identities to do the following. 1. Evaluate trigonometric functions. 2. Simplify trigonometric expressions.

Introduction

Introduction cont’d

Example 2 – Simplifying a Trigonometric Expression Simplify sin x cos2 x – sin x. Solution: First factor out a common monomial factor and then use a fundamental identity. sin x cos2 x – sin x = sin x (cos2 x – 1) = –sin x(1 – cos2 x) = –sin x(sin2 x) = –sin3 x Factor out common monomial factor. Factor out –1. Pythagorean identity. Multiply.

Example 3 – Factoring Trigonometric Expressions

Example 4 – Factoring Trigonometric Expressions

Example 5 – Simplifying Trigonometric Expressions

Example 6 – Adding Trigonometric Expressions

Example 7 – Rewriting a Trigonometric Expression Rewrite so that it is not in fractional form. Solution: From the Pythagorean identity cos2 x = 1 – sin2 x = (1 – sin x)(1 + sin x), you can see that multiplying both the numerator and the denominator by (1 – sin x) will produce a monomial denominator. Multiply numerator and denominator by (1 – sin x). Multiply.

Example 7 – Solution cont’d Pythagorean identity. Write as separate fractions. Product of fractions. Reciprocal and quotient identities.

Example 8 – Trigonometric Substitution

Example 9 – Rewriting a Logarithmic Expression

What You Should Learn Verify trigonometric identities.

Verifying Trigonometric Identities

Example 1 – Verifying a Trigonometric Identity Verify the identity (sec2  – 1) / (sec2  ) = sin2 . Solution: Pythagorean identity Simplify. Reciprocal identity Quotient identity Simplify.

Example 1 – Solution cont’d Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side. Simplify.

Example 2 – Verifying a Trigonometric Identity

Example 3 – Verifying a Trigonometric Identity

Example 4 – Converting to Sines and Cosines

Example 5 – Verifying a Trigonometric Identity

Example 6 – Verifying a Trigonometric Identity

Example 7 – Verifying a Trigonometric Identity

Multiple-Angle Formulas

What You Should Learn Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions.

Multiple-Angle Formulas

Example 1 – Solving a Multiple-Angle Equation Solve 2 cos x + sin 2x = 0. Solution: Begin by rewriting the equation so that it involves functions of x (rather than 2x). Then factor and solve. 2 cos x + sin 2x = 0 2 cos x + 2 sin x cos x = 0 2 cos x(1 + sin x) = 0 2 cos x = 0 and 1 + sin x = 0 Write original equation. Double-angle formula. Factor. Set factors equal to zero.

Example 1 – Solution So, the general solution is and cont’d So, the general solution is and where n is an integer. Try verifying these solutions graphically. Solutions in [0, 2)

Half-Angle Formulas

Half-Angle Formulas

Example 6 – Using a Half-Angle Formula Find the exact value of sin 105. Solution: Begin by noting that 105 is half of 210. Then, using the half-angle formula for sin(u / 2) and the fact that 105 lies in Quadrant II, you have

Example 6 – Solution cont’d . The positive square root is chosen because sin  is positive in Quadrant II.

Product-to-Sum Formulas

Product-to-Sum Formulas

Example 8 – Writing Products as Sums Rewrite the product cos 5x sin 4x as a sum or difference. Solution: Using the appropriate product-to-sum formula, you obtain cos 5x sin 4x = [sin(5x + 4x) – sin(5x – 4x)] = sin 9x – sin x.