Unit 6 Introduction to Trigonometry The Unit Circle (Unit 6.3) - - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry The Unit Circle (Unit 6.3)

William (Bill) Finch

Mathematics Department Denton High School

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

• W. Finch

DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

• W. Finch

DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Lesson Goals

When you have completed this lesson you will:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

• W. Finch

DHS Math Dept Unit Circle 2 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Trigonometric Functions of Any Angle

θ is an angle in standard position, P(x, y) is a point on the terminal side, and r is the distance from P to the origin (denominators = 0): sin θ = y r csc θ = r y cos θ = x r sec θ = r x tan θ = y x cot θ = x y x y r P(x, y) θ Recall the equation of a circle centered at the

• rigin:

r 2 = x2 + y 2 r =

• x2 + y 2
• W. Finch

DHS Math Dept Unit Circle 3 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 1

The point (−3, 2) is on the terminal side of an angle in standard position. Find the exact values of the six trigonometric functions of θ.

• W. Finch

DHS Math Dept Unit Circle 4 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Quadrental Angles

Quadrental angles terminate on an axis.

x y θ (r, 0)

θ = 0◦ or 0 radians

x y (0, r) θ

θ = 90◦ or π/2 radians

x y (−r, 0) θ

θ = 180◦ or π radians

x y (0, −r) θ

θ = 270◦ or 3π/2 radians

• W. Finch

DHS Math Dept Unit Circle 5 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 2

Find the exact value of each trigonometric function, if defined. If not defined, write undefined. a) cos π b) tan(−270◦) c) sec 3π 2 d) sin 5π

• W. Finch

DHS Math Dept Unit Circle 6 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Angles Not Acute or Quadrental

Quadrant II x y r (−a, b) b a θ θ′ sin θ = b r sin θ′ = b r cos θ = −a r cos θ′ = a r tan θ = −b a tan θ′ = b a

• W. Finch

DHS Math Dept Unit Circle 7 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Angles Not Acute or Quadrental

Quadrant III x y r (−a, −b) b a θ θ′ sin θ = −b r sin θ′ = b r cos θ = −a r cos θ′ = a r tan θ = b a tan θ′ = b a

• W. Finch

DHS Math Dept Unit Circle 8 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Angles Not Acute or Quadrental

Quadrant IV x y r (a, −b) b a θ θ′ sin θ = −b r sin θ′ = b r cos θ = a r cos θ′ = a r tan θ = −b a tan θ′ = b a

• W. Finch

DHS Math Dept Unit Circle 9 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Reference Angles

If θ is an angle in standard position, its reference angle θ′ is the acute angle formed by the terminal side of θ and the x-axis.

x y θ

θ′ = θ

x y θ θ′

θ′ = 180◦ − θ θ′ = π − θ

x y θ θ′

θ′ = θ − 180◦ θ′ = θ − π

x y θ θ′

θ′ = 360◦ − θ θ′ = 2π − θ

• W. Finch

DHS Math Dept Unit Circle 10 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 3

Sketch the angle and the identify its reference angle. a) −150◦ b) 315◦ c) 3π 4 d) 5π 3

• W. Finch

DHS Math Dept Unit Circle 11 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Evaluating Trigonometric Functions of Any Angle

• 1. Sketch the angle.
• 2. Determine the reference angle θ′.
• 3. Find the value of the trig

function for θ′.

• 4. Determine the sign (pos or neg)

based on the quadrant containing the terminal side of θ. x y

Quad I sin θ : + cos θ : + tan θ : + Quad II sin θ : + cos θ : − tan θ : − Quad III sin θ : − cos θ : − tan θ : + Quad IV sin θ : − cos θ : + tan θ : −

• W. Finch

DHS Math Dept Unit Circle 12 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Special Reference Angles

θ (radians) π 6 π 4 π 3 θ (degrees) 30◦ 45◦ 60◦ sin θ 1 2 √ 2 2 √ 3 2 cos θ √ 3 2 √ 2 2 1 2 tan θ √ 3 3 1 √ 3

• W. Finch

DHS Math Dept Unit Circle 13 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 4

Find the exact value of each expression. a) sin 4π 3 b) sec 15π 4 c) tan 150◦ d) cos (−120◦)

• W. Finch

DHS Math Dept Unit Circle 14 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 5

Let sec θ = √ 29 5 , where sin θ > 0. Find the exact values of the remaining five trigonometric functions of θ.

• W. Finch

DHS Math Dept Unit Circle 15 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

The Unit Circle

A unit circle is a circle of radius 1 centered at the origin. The radian measure of a central angle is θ = s r = s 1 = s so the arc length intercepted by θ equals the angle’s radian measure. x y r r s θ (1, 0) (0, 1) (−1, 0) (0, −1)

• W. Finch

DHS Math Dept Unit Circle 16 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

The Unit Circle and the Wrapping Function

Place a number line vertically tangent to a unit circle at (1, 0). Wrap this line around the circle (counterclockwise for positive values and clockwise for negative values), each point t on the line would map to a unique point P(x, y) on the circle. This is referred to as the wrapping function w(t). Since r = 1, the six trigonometric rations of angle t can be defined in terms of just x and y. x y 1 t (1, 0) P(x, y) t

• W. Finch

DHS Math Dept Unit Circle 17 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Trigonometric Functions on the Unit Circle

sin t = y csc t = 1 y cos t = x sec t = 1 x tan t = y x cot t = x y And, of course, no denominator = 0. x y 1 t t P(x, y) P(cos t, sin t) These functions are referred to as circular functions

• W. Finch

DHS Math Dept Unit Circle 18 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

16-Point Unit Circle

x y 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦ 360◦ 45◦ 135◦ 225◦ 315◦

π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6

π

7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6

2π (1, 0) √

3 2 , 1 2

• √

2 2 , √ 2 2

• 1

2, √ 3 2

• (0, 1)
• − 1

2, √ 3 2

• −

√ 2 2 , √ 2 2

• −

√ 3 2 , 1 2

• (−1, 0)
• −

√ 3 2 , − 1 2

• −

√ 2 2 , − √ 2 2

• − 1

2, − √ 3 2

• (0, −1)
• 1

2, − √ 3 2

• √

2 2 , − √ 2 2

• √

3 2 , − 1 2

• W. Finch

DHS Math Dept Unit Circle 19 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

16-Point Unit Circle

x y 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦ 360◦ 45◦ 135◦ 225◦ 315◦

π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6

π

7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6

2π (1, 0) √

3 2 , 1 2

• √

2 2 , √ 2 2

• 1

2, √ 3 2

• (0, 1)
• − 1

2, √ 3 2

• −

√ 2 2 , √ 2 2

• −

√ 3 2 , 1 2

• (−1, 0)
• −

√ 3 2 , − 1 2

• −

√ 2 2 , − √ 2 2

• − 1

2, − √ 3 2

• (0, −1)
• 1

2, − √ 3 2

• √

2 2 , − √ 2 2

• √

3 2 , − 1 2

• 30◦

60◦

• 45◦

π 6 π 4 π 3

(1, 0) √

3 2 , 1 2

• √

2 2 , √ 2 2

• 1

2, √ 3 2

• (0, 1)

x (1, 0)

• W. Finch

DHS Math Dept Unit Circle 20 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 6

Find the exact value of each expression. a) sin 7π 6 b) cos π 3 c) tan 4π 3 d) sec 270◦

• W. Finch

DHS Math Dept Unit Circle 21 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Periodic Functions

A number line can be wrapped around a circle infinitely many times, so the domain of both the sine and cosine functions is (−∞, ∞). This means more than one value t will be mapped

• nto the same point P(x, y). Graphing ordered pairs of the

form (t, sin t) shows how the function repeats periodically.

• W. Finch

DHS Math Dept Unit Circle 22 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Periodic Functions

A function y = f (t) is periodic if there exists a positive real number c such that f (t + c) = f (t) for all values of t in the domain of f . The smallest number c for which f is periodic is called the period of f . sin(t + n · 2π) = sin t period = 2π cos(t + n · 2π) = cos t period = 2π tan(t + n · π) = tan t period = π

• W. Finch

DHS Math Dept Unit Circle 23 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

Example 7

Use the period of each function to determine an exact value. a) cos 9π 4 b) sin −2π 3

• c) tan 29π

6

• W. Finch

DHS Math Dept Unit Circle 24 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

◮ Do problems Chap 4.3 #1, 5, 9-31 odd, , 33-37 odd,

43-57 odd, 61-65 odd, 73, 75

• W. Finch

DHS Math Dept Unit Circle 25 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

◮ Do problems Chap 4.3 #1, 5, 9-31 odd, , 33-37 odd,

43-57 odd, 61-65 odd, 73, 75

• W. Finch

DHS Math Dept Unit Circle 25 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

◮ Do problems Chap 4.3 #1, 5, 9-31 odd, , 33-37 odd,

43-57 odd, 61-65 odd, 73, 75

• W. Finch

DHS Math Dept Unit Circle 25 / 25

Introduction Trig Functions – Circle Quadrental Angles Other Angles Unit Circle Periodic Functions Summary

What You Learned

You can now:

◮ Find values of trigonometric functions for any angle. ◮ Find the values of trigonometric functions using the unit

circle.

◮ Do problems Chap 4.3 #1, 5, 9-31 odd, , 33-37 odd,

43-57 odd, 61-65 odd, 73, 75

• W. Finch

DHS Math Dept Unit Circle 25 / 25