Warm Up

Warm Up

1 The hypotenuse of a 45◦ − 45◦ − 90◦ triangle is 1 unit in length.What is the measure of each of the other two sides of the triangle?

2 The hypotenuse of a 30◦ − 60◦ − 90◦ triangle is 1 unit in length.What is the measure of each of the other two sides of the triangle?

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 1 / 27

4.2 – Trigonometric Functions: The Unit Circle

Pre-Calculus

Mr. Niedert

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 2 / 27

4.2 – Trigonometric Functions: The Unit Circle

1 The Unit Circle

2 Trigonometric Functions

3 Domain and Period of Sine and Cosine

4 Evaluating Trigonometric Functions with a Calculator

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27

4.2 – Trigonometric Functions: The Unit Circle

1 The Unit Circle

2 Trigonometric Functions

3 Domain and Period of Sine and Cosine

4 Evaluating Trigonometric Functions with a Calculator

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27

4.2 – Trigonometric Functions: The Unit Circle

1 The Unit Circle

2 Trigonometric Functions

3 Domain and Period of Sine and Cosine

4 Evaluating Trigonometric Functions with a Calculator

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27

4.2 – Trigonometric Functions: The Unit Circle

1 The Unit Circle

2 Trigonometric Functions

3 Domain and Period of Sine and Cosine

4 Evaluating Trigonometric Functions with a Calculator

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27

Today’s Learning Target(s)

1 I can use 45◦ − 45◦ − 90◦ triangles and 30◦ − 60◦ − 90◦ triangles tofind the coordinates of various points on the unit circle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 4 / 27

The Unit Circle

Using what we found with the 45◦ − 45◦ − 90◦ triangles and the30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as theunit circle.

You will need to know the unit circle like the back of your handthrough the remainder of this year.

I also would like for you to understand where it comes from asopposed to simply memorizing the unit circle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 5 / 27

The Unit Circle

Using what we found with the 45◦ − 45◦ − 90◦ triangles and the30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as theunit circle.

You will need to know the unit circle like the back of your handthrough the remainder of this year.

I also would like for you to understand where it comes from asopposed to simply memorizing the unit circle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 5 / 27

The Unit Circle

Using what we found with the 45◦ − 45◦ − 90◦ triangles and the30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as theunit circle.

You will need to know the unit circle like the back of your handthrough the remainder of this year.

I also would like for you to understand where it comes from asopposed to simply memorizing the unit circle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 5 / 27

The Unit Circle

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 6 / 27

4.2 – Trigonometric Functions: The Unit Circle QuizTomorrow

You will be given a blank unit circle and be expected to complete the unitcircle tomorrow as your warm up.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 7 / 27

Today’s Learning Target(s)

1 I can use the unit circle to evaluate sine, cosine, and tangent forangles in degrees and radians.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 8 / 27

Sine, Cosine, and Tangent

Sine, Cosine, and Tangent

Trigonometric functions can be defined in terms of their (x , y)coordinates. Let t be real number and let (x , y) be the point on the unitcircle corresponding to t.

sin t = y cos t = x tan t =y

x, x 6= 0

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 9 / 27

Evaluating Sine, Cosine, and Tangent

Example

Evaluate each of the following.

a sin 3π4

b cos 300◦

c tan 3π2

d tan π6

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 10 / 27

Sine, Cosine, and Tangent Domino Activity

1 You will receive a set of dominoes.

2 You need to connect these dominoes to one another so that the valueon the left of the domino is equal to the value of the trigonometricfunction on the right of the previous domino.

3 I will come to check on these as you are working.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 11 / 27

Warm Up

Warm Up

Find the sine, cosine, and tangent at t = π4 .

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 12 / 27

ACT Incentives

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 13 / 27

Today’s Learning Target(s)

1 I can evaluate trigonometric functions with and without a calculator.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 14 / 27

The Six Trigonometric Functions

The Six Trigonometric Functions

The six trigonometric functions can be defined in terms of their (x , y)coordinates. Let t be real number and let (x , y) be the point on the unitcircle corresponding to t.

sin t = y cos t = x tan t =y

x, x 6= 0

csc t =1

y, y 6= 0 sec t =

1

x, x 6= 0 cot t =

x

y, y 6= 0

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 15 / 27

Evaluating Trigonometric Functions

Example

Evaluate the six trigonometric functions for t = 2π3

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 16 / 27

Evaluating Trigonometric Functions

Practice

Evaluate the six trigonometric functions at t = −π6 .

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 17 / 27

Domain and Range of Sine and Cosine

Demonstration #1

What is the domain of y = sin x?

What is the range?

Demonstration #2

What is the domain of y = cos x? What is the range?

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27

Domain and Range of Sine and Cosine

Demonstration #1

What is the domain of y = sin x? What is the range?

Demonstration #2

What is the domain of y = cos x? What is the range?

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27

Domain and Range of Sine and Cosine

Demonstration #1

What is the domain of y = sin x? What is the range?

Demonstration #2

What is the domain of y = cos x?

What is the range?

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27

Domain and Range of Sine and Cosine

Demonstration #1

What is the domain of y = sin x? What is the range?

Demonstration #2

What is the domain of y = cos x? What is the range?

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27

Periodic Functions

As we saw on the previous slide, we have a domain of all realnumbers, but the y -values repeat over and over again.

Functions that behave in such a repetitive (or cyclic) manner arecalled periodic.

The period of the function refers to how “long” it takes for they -values to complete a full cycle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 19 / 27

Periodic Functions

As we saw on the previous slide, we have a domain of all realnumbers, but the y -values repeat over and over again.

Functions that behave in such a repetitive (or cyclic) manner arecalled periodic.

The period of the function refers to how “long” it takes for they -values to complete a full cycle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 19 / 27

Periodic Functions

As we saw on the previous slide, we have a domain of all realnumbers, but the y -values repeat over and over again.

Functions that behave in such a repetitive (or cyclic) manner arecalled periodic.

The period of the function refers to how “long” it takes for they -values to complete a full cycle.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 19 / 27

Even and Odd Trigonometric Functions

Back at the beginning of the year, we discussed that a function iseven if f (−x) = f (x).

Similarly, a function is odd if f (−x) = −f (x).

Even and Odd Trigonometric Functions

The cosine and secant functions are even.

cos(−t) = cos t sec(−t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

sin(−t) = − sin t csc(−t) = − csc ttan(−t) = − tan t cot(−t) = − cot t

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 20 / 27

Even and Odd Trigonometric Functions

Back at the beginning of the year, we discussed that a function iseven if f (−x) = f (x).

Similarly, a function is odd if f (−x) = −f (x).

Even and Odd Trigonometric Functions

The cosine and secant functions are even.

cos(−t) = cos t sec(−t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

sin(−t) = − sin t csc(−t) = − csc ttan(−t) = − tan t cot(−t) = − cot t

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 20 / 27

Even and Odd Trigonometric Functions

Back at the beginning of the year, we discussed that a function iseven if f (−x) = f (x).

Similarly, a function is odd if f (−x) = −f (x).

Even and Odd Trigonometric Functions

The cosine and secant functions are even.

cos(−t) = cos t sec(−t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

sin(−t) = − sin t csc(−t) = − csc ttan(−t) = − tan t cot(−t) = − cot t

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 20 / 27

Using the Period to Evaluate the Sine and Cosine

Example

Find the following.

a cos 9π3

b sin(−11π

2

)c If sin(t) = 4

5 , find sin(−t).

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 21 / 27

Using the Period to Evaluate the Sine and Cosine

Practice

Find the following.

a sin 13π6

b cos(−11π

6

)c If tan(t) = 2

3 , find tan(−t).

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 22 / 27

Using a Calculator to Evaluate Trigonometric Functions

Walk-Through

Evaluate each of the following using a calculator.

a cos 2π3

b sin 5π7

c csc 2

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 23 / 27

4.2 – Trigonometric Functions: The Unit Circle Assignment

4.2 – Trigonometric Functions: The Unit Circle Assignmentpg. 299-300 #6-52 even

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 24 / 27

Warm Up

Warm Up

Evaluate all six trigonometric functions (sine, cosine, tangent, cosecant,

secant, and cotangent) at t =7π

6.

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 25 / 27

Bell Schedule

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 26 / 27

4.2 – Trigonometric Functions: The Unit Circle Assignment

4.2 – Trigonometric Functions: The Unit Circle Assignmentpg. 299-300 #6-52 even

Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 27 / 27