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).