Math 2204

Unit 4: Trigonometric equations

Section 4.1

Trigonometric Equation

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

DIRECTION ON THE UNIT CIRCLE

COORDINATES ON UNIT CIRCLE

USING TRIG CIRCLE COORDINATES TO SIMPLIFY EXPRESSIONS

USING TRIG CIRCLE COORDINATES TO SIMPLIFY EXPRESSIONS

USING TRIG CIRCLE COORDINATES TO SIMPLIFY EXPRESSIONS

EXAMPLE 1Find the exact value of the sine and cosine of 300o.

Solution1. Sketch a diagram showing a

rotation of 300o

2. The side opposite 30o is the x-coordinate, which is , and this gives us the cosine of the angle.

3. The side opposite 60o 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:

EXAMPLE 2Find the exact value of sine and cosine of -225o

Solution1. 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 45o - 45o - 90o so both sides are

Since the point is in the second quadrant, the x-coordinate is negative and the y-coordinate is positive.

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.

SOLVING TRIGONOMETRIC EQUATIONS USING EXACT VALUES

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

SOLVING TWO EQUATIONS TOGETHER

SOLVING TWO EQUATIONS TOGETHER GRAPHICALLY

SOLVING TWO EQUATIONS TOGETHER ALGEBRAICALLY

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.

Section 4.2

Trigonometric identities

CURRICULUM OUTCOMES CONTAINED IN SECTION 4.2

A1demonstrate an understanding of irrational numbers in applications

B1 demonstrate an understanding of the relationship between operations on fractions and rational algebraic expressions

B4 use the calculator correctly and efficientlyC9 analyze tables and graphs of various sine and cosine functions to

find patterns, identify characteristics and determine equationsC24derive and apply the reciprocal and Pythagorean identitiesC25prove trigonometric identitiesC28analyze and solve trigonometric equations with and without

technology

SUMMARY OF TRIG FUNCTIONS, EQUATIONS AND IDENTITIES

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

EXAMPLE 1

EXAMPLE 2

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.

EXAMPLE 1

EXAMPLE 2

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.

EXAMPLE 1

EXAMPLE 2

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

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.

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

Section 4.3

Radian measure

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 relationshipsD2demonstrate an understanding of the connection between

degree and radian measure and apply them

COMPARISON OF DEGREE AND RADIAN MEASURE

EXAMPLE: DEGREES TO RADIANS

EXAMPLE: RADIANS TO DEGREES

RADIAN TO DEGREES AND DEGREES TO RADIANS

RADIANS AND DEGREES

SOLVING EQUATIONS IN TRIGONOMETRY

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.

assignment

End of unit 4