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 2 Trigonometric Functions
3 Domain and Period of Sine and Cosine
4 Evaluating Trigonometric Functions with a Calculator
4.2 – Trigonometric Functions: The Unit Circle
1 The Unit Circle
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27 3 Domain and Period of Sine and Cosine
4 Evaluating Trigonometric Functions with a Calculator
4.2 – Trigonometric Functions: The Unit Circle
1 The Unit Circle
2 Trigonometric Functions
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 3 / 27 4 Evaluating Trigonometric Functions with a Calculator
4.2 – Trigonometric Functions: The Unit Circle
1 The Unit Circle
2 Trigonometric Functions
3 Domain and Period of Sine and Cosine
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 to find the coordinates of various points on the unit circle.
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 4 / 27 You will need to know the unit circle like the back of your hand through the remainder of this year. I also would like for you to understand where it comes from as opposed to simply memorizing the unit circle.
The Unit Circle
Using what we found with the 45◦ − 45◦ − 90◦ triangles and the 30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as the unit circle.
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 5 / 27 I also would like for you to understand where it comes from as opposed to simply memorizing the unit circle.
The Unit Circle
Using what we found with the 45◦ − 45◦ − 90◦ triangles and the 30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as the unit circle. You will need to know the unit circle like the back of your hand through the remainder of this year.
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 the 30◦ − 60◦ − 90◦ triangles, we can complete what is referred to as the unit circle. You will need to know the unit circle like the back of your hand through the remainder of this year. I also would like for you to understand where it comes from as opposed 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 Quiz Tomorrow
You will be given a blank unit circle and be expected to complete the unit circle 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 for angles 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 unit circle corresponding to t. y sin t = y cos t = x tan t = , x 6= 0 x
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 9 / 27 Evaluating Sine, Cosine, and Tangent
Example Evaluate each of the following. 3π a sin 4 b cos 300◦ 3π c tan 2 π d tan 6
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 value on the left of the domino is equal to the value of the trigonometric function 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 π Find the sine, cosine, and tangent at t = 4 .
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 unit circle corresponding to t. y sin t = y cos t = x tan t = , x 6= 0 x 1 1 x csc t = , y 6= 0 sec t = , x 6= 0 cot t = , y 6= 0 y x y
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 15 / 27 Evaluating Trigonometric Functions
Example 2π Evaluate the six trigonometric functions for t = 3
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 16 / 27 Evaluating Trigonometric Functions
Practice π Evaluate the six trigonometric functions at t = − 6 .
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 17 / 27 What is the range?
Demonstration #2 What is the domain of y = cos x? What is the range?
Domain and Range of Sine and Cosine
Demonstration #1 What is the domain of y = sin x?
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27 Demonstration #2 What is the domain of y = cos x? What is the range?
Domain and Range of Sine and Cosine
Demonstration #1 What is the domain of y = sin x? What is the range?
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 18 / 27 What is the range?
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?
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 Functions that behave in such a repetitive (or cyclic) manner are called periodic. The period of the function refers to how “long” it takes for the y-values to complete a full cycle.
Periodic Functions
As we saw on the previous slide, we have a domain of all real numbers, but the y-values repeat over and over again.
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 19 / 27 The period of the function refers to how “long” it takes for the y-values to complete a full cycle.
Periodic Functions
As we saw on the previous slide, we have a domain of all real numbers, but the y-values repeat over and over again. Functions that behave in such a repetitive (or cyclic) manner are called periodic.
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 real numbers, but the y-values repeat over and over again. Functions that behave in such a repetitive (or cyclic) manner are called periodic. The period of the function refers to how “long” it takes for the y-values to complete a full cycle.
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 19 / 27 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 t tan(−t) = − tan t cot(−t) = − cot t
Even and Odd Trigonometric Functions
Back at the beginning of the year, we discussed that a function is even if f (−x) = f (x).
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 20 / 27 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 t tan(−t) = − tan t cot(−t) = − cot t
Even and Odd Trigonometric Functions
Back at the beginning of the year, we discussed that a function is even if f (−x) = f (x). Similarly, a function is odd if f (−x) = −f (x).
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 is even 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 t tan(−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. 9π a cos 3 −11π b sin 2 4 c If sin(t) = 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. 13π a sin 6 −11π b cos 6 2 c If tan(t) = 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. 2π a cos 3 5π b sin 7 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 Assignment pg. 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, 7π secant, and cotangent) at t = . 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 Assignment pg. 299-300 #6-52 even
Pre-Calculus 4.2 – Trig Func: The Unit Circle Mr. Niedert 27 / 27