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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
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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
Example 1 Find the exact value of the sine and cosine of 300o. Solution • Sketch a diagram showing a rotation of300o • The side opposite 30o is the x-coordinate, which is ,and this gives us the cosine of theangle. • 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 2 Find the exact value of sine and cosine of -225o Solution • Remember that for a negative angle the rotation is clockwise. • Sketch a diagram showing the angle and the corresponding right triangle as on the right. • 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 non-exact values (using calculator)
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 A1 demonstrate an understanding of irrational numbers in applications B1 demonstrate 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
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
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.
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.
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 • select the expressions on one side of the equal sign to work with, usually start with the left hand side. • write all the expressions on that side in terms of sine or cosine and/or apply the various trigonometric identities that you have memorized. • simplify using algebraic manipulation, which may include factoring. • 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 A1 demonstrate an understanding of irrational numbers in applications B4 use 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
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.
