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Math 2204 Unit 4: Trigonometric equations

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Section 4.1 Trigonometric Equation

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Curriculum outcomes covered in section 4.1 A1 demonstrate an understanding of irrational numbers in applications B4 use the calculator correctly and efficiently C1 model situations with sinusoidal functions C9 analyze tables and graphs of various sine and cosine functions to find patterns, identify characteristics and determine equations C15 demonstrate an understanding of sine and cosine ratios and functions for non-acute angles C18 interpolate and extrapolate to solve problems C27 apply function notation to trigonometric equations C28 analyze and solve trigonometric equations with and without technology C30 Demonstrate an understanding of the relationship between solving algebraic and trigonometric equations

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DIRECTION ON THE UNIT CIRCLE

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COORDINATES ON UNIT CIRCLE

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USING TRIG CIRCLE COORDINATES TO SIMPLIFY EXPRESSIONS

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EXAMPLE 1 Find the exact value of the sine and cosine of 300 o. Solution 1.Sketch a diagram showing a rotation of 300 o 2.The side opposite 30 o is the x- coordinate, which is, and this gives us the cosine of the angle. 3.The side opposite 60 o is the y- coordinate which is and this gives us the sine of the angle. Since the point is in the fourth quadrant the x-coordinate is positive and the y-coordinate is negative. Thus we can say:

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EXAMPLE 2 Find the exact value of sine and cosine of -225 o Solution 1.Remember that for a negative angle the rotation is clockwise. 2.Sketch a diagram showing the angle and the corresponding right triangle as on the right. 3.The resulting triangle is 45 o - 45 o - 90 o so both sides are Since the point is in the second quadrant, the x-coordinate is negative and the y-coordinate is positive.

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PROBLEMS DONE ON BOARD (IF YOU ARE ABSENT OR REFUSE TO WRITE THESE SOLUTIONS DOWN IT IS UP TO YOU TO GET THE NOTES FROM A CLASSMATE) Do CYU Questions 7-9,13,16,18 on pages 135 - 138.

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SOLVING TRIGONOMETRIC EQUATIONS USING EXACT VALUES

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SOLVING TRIGONOMETRIC EQUATIONS USING NON-EXACT VALUES (USING CALCULATOR)

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SOLVING TWO EQUATIONS TOGETHER

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SOLVING TWO EQUATIONS TOGETHER GRAPHICALLY

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SOLVING TWO EQUATIONS TOGETHER ALGEBRAICALLY

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PROBLEMS DONE ON BOARD (IF YOU ARE ABSENT OR REFUSE TO WRITE THESE SOLUTIONS DOWN IT IS UP TO YOU TO GET THE NOTES FROM A CLASSMATE) Do CYU Questions 28, 30,31 on pages 142 - 145.

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Section 4.2 Trigonometric identities

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CURRICULUM OUTCOMES CONTAINED IN SECTION 4.2 A1demonstrate an understanding of irrational numbers in applications B1demonstrate an understanding of the relationship between operations on fractions and rational algebraic expressions B4use the calculator correctly and efficiently C9analyze tables and graphs of various sine and cosine functions to find patterns, identify characteristics and determine equations C24derive and apply the reciprocal and Pythagorean identities C25prove trigonometric identities C28analyze and solve trigonometric equations with and without technology

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SUMMARY OF TRIG FUNCTIONS, EQUATIONS AND IDENTITIES

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SIMPLIFYING RATIONAL EXPRESSIONS To simplify a rational expression (as you did with fractions) you remove the common factors from its numerator and denominator. This is best explained by use of an example

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EXAMPLE 1

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EXAMPLE 2

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MULTIPLYING & DIVIDING To multiply rational expressions, multiply together the numerators and denominators, factor, and remove common factors from the numerator and denominator. Division is identical except that you first have to change the division to a multiplication.

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EXAMPLE 1

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EXAMPLE 2

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ADDITION & SUBTRACTION As you learned with your work on fractions in earlier grades, to add and subtract rational expressions requires that they have a common denominator. Once two expressions have a common denominator, we simply add or subtract their numerators. It is easiest if we find the least common denominator of the two fractions before we start to add or subtract. To do this, factor the denominator of all fractions.

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EXAMPLE 1

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EXAMPLE 2

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PROBLEMS DONE ON BOARD (IF YOU ARE ABSENT OR REFUSE TO WRITE THESE SOLUTIONS DOWN IT IS UP TO YOU TO GET THE NOTES FROM A CLASSMATE) Do CYU Questions pg. 155 and 156 #’s 15, 16, 18, 19

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A GOOD STRATEGY TO USE IN VERIFYING OR PROVING IDENTITIES IS 1.select the expressions on one side of the equal sign to work with, usually start with the left hand side. 2.write all the expressions on that side in terms of sine or cosine and/or apply the various trigonometric identities that you have memorized. 3.simplify using algebraic manipulation, which may include factoring. 4.if necessary, repeat the process for the other side of the equal sign. Hopefully, if you have done your work correctly, both sides of the equal sign will give the same expression or value, regardless of the measure of the angle.

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PROBLEMS DONE ON BOARD (IF YOU ARE ABSENT OR REFUSE TO WRITE THESE SOLUTIONS DOWN IT IS UP TO YOU TO GET THE NOTES FROM A CLASSMATE) Do CYU Questions 11, 12,13,14,17,19

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Section 4.3 Radian measure

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CURRICULUM OUTCOMES COVERED IN SECTION 4.3 A1demonstrate an understanding of irrational numbers in applications B4use the calculator correctly and efficiently C9analyze tables and graphs of various sine and cosine functions to find patterns, identify characteristics and determine equations D1derive, analyze, and apply angle and arc length relationships D2demonstrate an understanding of the connection between degree and radian measure and apply them

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COMPARISON OF DEGREE AND RADIAN MEASURE

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EXAMPLE: DEGREES TO RADIANS

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EXAMPLE: RADIANS TO DEGREES

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RADIAN TO DEGREES AND DEGREES TO RADIANS

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RADIANS AND DEGREES

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SOLVING EQUATIONS IN TRIGONOMETRY

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PROBLEMS DONE ON BOARD (IF YOU ARE ABSENT OR REFUSE TO WRITE THESE SOLUTIONS DOWN IT IS UP TO YOU TO GET THE NOTES FROM A CLASSMATE) Do the CYU questions 6 - 22.

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assignment

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End of unit 4

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