Chapter 6: Trigonometry Section 6.4: Trigonometric Functions

Day 1

Trigonometric Ratios P (x.y) 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: r y θ x

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 Form an angle of t radians Begin with a Number t Determine sint, cost, tant

Trigonometric Ratios (x.y) 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: r y t x

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 -2y+5x=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 3y-4x=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 In the unit circle, the radius is always 1. So if r = 1, then: 1 1 -1 1 -1

Domain and Range sin and cos: Domain is the set of all real numbers! Range is the set of all real numbers between -1 and 1. tan: 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 30o 45o 60o Sin t 1/2 √2/2 √3/2 Cos t Tan t √3/3 1 √3 Csc t 2 √2 2√3/3 Sec t Cot t Cos 90o 60o 2 45o 30o 0o 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 30o. Cos 90o 60o 2 45o 30o 0o 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 45o. Cos 90o 60o 2 45o 30o 0o 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. 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 x- axis. t t 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 30o. π/6 1/2 √3/2 √3/3 2 2√3/3 √3 45o, π/4 √2/2 1 √2 60o, π/3 90o, π/2 Undef

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