Chapter 5 Trigonometric Functions of Angles

Chapter 5 Trigonometric Functions of Angles

Section 3 Points on Circles Using Sine and Cosine

Signs

Signs I

Signs (+, +) I

Signs II (+, +) I

Signs II (, +) (+, +) I

Signs II (, +) (+, +) I III

Signs II (, +) (+, +) I III (, )

Signs II (, +) (+, +) I III (, ) IV

Signs II (, +) (+, +) I III (, ) (+, ) IV

Circles and Trig What was the equation of the unit circle?

Circles and Trig What was the equation of the unit circle? x + y = 1

Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ).

Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ). Does anyone remember what this is called when we make this substitution?

Circles and Trig What was the equation of the unit circle? x + y = 1 What if we let x = cos(θ) and y = sin(θ). Does anyone remember what this is called when we make this substitution? The Pythagorean Identity cos (θ) + sin (θ) = 1

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ).

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) + 1 9 = 1

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) + 1 9 = 1 cos (θ) = 8 9

Using the Pythagorean Identity If sin(θ) = 1 3, find cos(θ). cos (θ) + sin (θ) = 1 ( ) 1 cos (θ) + = 1 3 cos (θ) + 1 9 = 1 cos (θ) = 8 9 cos(θ) = 3

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5.

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = 1 5

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = 1 5 4 5 + sin (θ) = 1

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = 1 5 4 5 + sin (θ) = 1 sin (θ) = 1 5

Using the Pythagorean Identity Find sin(θ) if cos(θ) = 5. cos (θ) + sin (θ) = 1 ( ) + sin (θ) = 1 5 4 5 + sin (θ) = 1 sin (θ) = 1 5 1 sin(θ) = 5

Visually Speaking... (x, y) 1 θ

Visually Speaking... (x, y) θ 1 cos(θ)

Visually Speaking... (x, y) 1 sin(θ) θ cos(θ)

The Relationships ( (, 3, 1 ( 1, 0) ( 1 ), 3 ) ) 5π 6 π π 3 3π 4 π (0, 1) π 3 π 4 π ( ) 1, 3 π 6 ( ), ( 3, 1 ) (1, 0) ( 3 (, 1, ) ( 1, ) 3 ) 7π 6 5π 4 4π 3 3π 7π 4 5π 3 (0, 1) 11π 6 ( 3 (,, ) 1 ( 1, ) 3 )

Using the Unit Circle Find sin ( ) 3π 4.

Using the Unit Circle Find sin ( 3π 4 ).

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ).

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1?

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6 What angle has a cosine of 3?

Using the Unit Circle Find sin ( 3π 4 ). Find cos(300 ). 1 What angle has a sine of 1? 7π 6, 11π 6 What angle has a cosine of 3? π 6, 11π 6

Using Reference Angles We don t need to memorize the whole circle - just the first quadrant.

Using Reference Angles We don t need to memorize the whole circle - just the first quadrant. Notice that all of the values around this circle are ± 1, ± and ± 3. The ± decision is based on what quadrant the reference angle lies in.

Using Reference Angles Find sin ( ) 7π 6.

Using Reference Angles Find sin ( 7π 6 ). 7π 6

Using Reference Angles Find sin ( 7π 6 ). 7π 6

Using Reference Angles Find sin ( 7π 6 ). 7π 6 π 6

Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6

Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6 (, )

Using Reference Angles Find sin ( 7π 6 ). ( 3, 1 ) 7π 6 π 6 (, ) sin ( ) 7π 6 = 1

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1.

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y?

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value?

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for?

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3.

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3. What is the corresponding angle in quadrant IV?

Using Reference Angles Find the angle(s) θ such that cos(θ) = 1. Does cos(θ) correspond to x or y? In what quadrants does cos(θ) have a positive value? In quadrant I, what angle are we looking for? π 3. What is the corresponding angle in quadrant IV? π 3, or 5π 3.

Different Sized Circles What if we have a different radius?

Different Sized Circles What if we have a different radius? Different Radii The coordinate of a point on a circle of radius r corresponding to an angle θ is given by (rcos(θ), rsin(θ)).

Different Sized Circles What if we have a different radius? Different Radii The coordinate of a point on a circle of radius r corresponding to an angle θ is given by (rcos(θ), rsin(θ)). This is the exact same thing, except we multiply the value in question is multiplied by the radius to find the value.