CHAPTER 6 TRIGONOMETRY Section 6 4 Trigonometric Functions

CHAPTER 6: TRIGONOMETRY Section 6. 4: Trigonometric Functions

DAY 1

TRIGONOMETRIC RATIOS Let θ be an angle in standard position and let P(x, y) be any point on the terminal side of θ. Let r be the distance from (x, y) to the origin: Then the trigonometric ratios of θ are defined as follows: P (x. y) r θ x y

TRIGONOMETRIC RATIOS Find Sin, Cos, and Tan of the angle θ, whose terminal side passes through the point (-3, -2). Find Sin, Cos, and Tan of the angle θ, whose terminal side passes through the point (-2, 3).

TRIGONOMETRIC FUNCTIONS Trigonometric ratios have been defined for all angles. But modern applications of trigonometry deal with functions whose domains consist of real numbers. The basic idea is quite simple: If t is a real number then: sin t is defined to be the sine of an angle of t radians; cos t is defined to be the cosine of an angle of t radians; and so on. Instead of starting with angles, as was done up until now, this new approach starts with a number and only then moves to angles Begin with a Number t Form an angle of t radians Determine sint, cost, tant

TRIGONOMETRIC RATIOS Let t be a real number. Choose any point (x, y) on the terminal side of an angle t radians in standard position. Then the trigonometric ratios of t radians are defined as follows: (x. y) r t x y

TRIGONOMETRIC RATIOS Find Sin t, Cos t, and Tan t when the terminal side of t radians passes through the point (5, -1). Find Sin t, Cos t, and Tan t when the terminal side of t radians passes through the point (-4, 4).

TRIGONOMETRIC RATIOS The terminal side of an angle of t radians lies in quadrant 1 on the line through the origin parallel to 2 y+5 x=12. Find Sin t, Cos t, and Tan t. YOU TRY!! The terminal side of an angle of t radians lies in quadrant 1 on the line through the origin parallel to 3 y-4 x=12. Find Sin t, Cos t, and Tan t.

HOMEWORK!!! Page 452: 1 -6 6. 4 Worksheet #1

DAY 2

TRIGONOMETRY AND THE UNIT CIRCLE 1 1 -1 In the unit circle, the radius is always 1. So if r = 1, then:

DOMAIN AND RANGE sin and cos: tan: Domain is the set of all real numbers! Range is the set of all real numbers between 1 and 1. Domain is the set of all real numbers except ±π/2 + kπ, where k = 0. ± 1, ± 2, … Range is the set of all real numbers!

EXACT VALUES OF OUR SPECIAL ANGLES Square root of finger over palm! t 30 o 45 o 60 o Sin t 1/2 √ 2/2 √ 3/2 Cos t √ 3/2 √ 2/2 1/2 Tan t √ 3/3 1 √ 3 Csc t 2 √ 2 2√ 3/3 Sec t 2√ 3/3 √ 2 2 Cot t √ 3 1 √ 3/3 Cos 90 o 60 o 45 o 30 o 2 0 o Sin Flip hand over for Tangent!!

EXACT VALUES OF OUR SPECIAL ANGLES Square root of finger over palm! Without using a calculator, Find the sin, cos, and tan of 30 o. Cos 90 o 60 o 45 o 30 o 2 0 o Sin Flip hand over for Tangent!!

EXACT VALUES OF OUR SPECIAL ANGLES Square root of finger over palm! Without using a calculator, Find the sin, cos, and tan of 45 o. Cos 90 o 60 o 45 o 30 o 2 0 o Sin Flip hand over for Tangent!!

TRIGONOMETRY AND THE UNIT CIRCLE Find sint, cost, and tant when the terminal side of an angle of t radians passes through the given point on the unit circle.

TRIG FUNCTION SIGNS

HOMEWORK!!! Page 452: 7 -10 Create a poster of Trig Signs in different quadrants. 50 pts � � � 1) Colorful 10 pts 2) All correct signs 30 pts 3) Neatness 10 pts

DAY 3

REFERENCE ANGLES Reference Angle is the positive acute angle formed by the terminal side of θ and the xaxis. t t t’=π-t t t t’=t-π t’=2π-t

REFERENCE ANGLES Find the reference angle to the given angle: Now find sin, cos, and tan for each problem and append the appropriate sign.

PRACTICE Find the exact value of the sin, cos, and tan of the number without using a calculator.

COMPLETE THE CHART T Sin Cos Tan Csc Sec Cot 30 o. π/6 1/2 √ 3/3 2 2√ 3/3 √ 3 45 o, π/4 √ 2/2 1 √ 2 1 60 o, π/3 √ 3/2 1/2 √ 3 2√ 3/3 2 √ 3/3 90 o, π/2 1 0 Undef 1 Undef 0

HOMEWORK!!! Pg. 452 11 -14, 15 -23 part a only, 24 -35. Complete the chart!

DAY 4

EVALUATING EXPRESSIONS Write the expression as a single real number.

EVALUATING EXPRESSIONS YOU TRY!!! Write the expression as a single real number.

HOMEWORK!! Pg. 452: 36 -53 6. 4 Worksheet #2