Special Angles and Reference Angles PDF

Special Angles and Reference Angles April 7, 2025 Special and Reference Angles April 7, 2025 1 / 16 Agenda for the Day You all solve the mystery of: “What are the sin and cos of 45◦ = π4 , 30◦ = π6 , and 60◦ = π3 ”? Learn what reference angles are! Fill o...

Special Angles and Reference Angles April 7, 2025 Special and Reference Angles April 7, 2025 1 / 16 Agenda for the Day You all solve the mystery of: “What are the sin and cos of 45◦ = π4 , 30◦ = π6 , and 60◦ = π3 ”? Learn what reference angles are! Fill out more of the unit circle sheet Special and Reference Angles April 7, 2025 2 / 16 Logistical Notes I’ve graded Challenge Problem 1; I’ll hand it back now! People did well proving the Triangle Postulate; proving the Pythagorean Theorem was trickier Remember: If you’re trying to prove a statement, you can’t use that statement in your proof! Remember: The Pythagorean Theorem ONLY applies to right triangles. (But the triangle inequality applies to all triangles.) If you weren’t here last week Wednesday, you can make up the Challenge Problem in office hours! The WeBWorKs from Week 1 will be due this week; make sure to get them done Though if you need an extension, just ask! Remember: If you’re stuck on a problem, you can use the “Email Instructor” button! Also remember the Canvas Quiz! It’s due tonight! Special and Reference Angles April 7, 2025 3 / 16 Exact Values of Trigonometric Functions Suppose you want to find, say, sin(28.6◦ ). Special and Reference Angles April 7, 2025 4 / 16 Exact Values of Trigonometric Functions Suppose you want to find, say, sin(28.6◦ ). If you’re fine with a decimal approximation of it, just plug it into your calculator! sin(28.6◦ ) = 0.47869... Special and Reference Angles April 7, 2025 4 / 16 Exact Values of Trigonometric Functions Suppose you want to find, say, sin(28.6◦ ). If you’re fine with a decimal approximation of it, just plug it into your calculator! sin(28.6◦ ) = 0.47869... But what if you want an exact value for it? No decimals, just a precise closed-form expression Well, unless you want some hideously massive expression where you take the 30th root of an imaginary number or something...... the best you can really do is “sin(28.6◦ )” Special and Reference Angles April 7, 2025 4 / 16 Exact Values of Trigonometric Functions Suppose you want to find, say, sin(28.6◦ ). If you’re fine with a decimal approximation of it, just plug it into your calculator! sin(28.6◦ ) = 0.47869... But what if you want an exact value for it? No decimals, just a precise closed-form expression Well, unless you want some hideously massive expression where you take the 30th root of an imaginary number or something...... the best you can really do is “sin(28.6◦ )” Thankfully, there are some angles where you can find exact values for their sine and cosine We’ve already computed the exact values of sin(0◦ ), cos(0◦ ), sin(90◦ ), cos(90◦ ), etc. We’re going to find the exact sines and cosines of 3 more angles today! Special and Reference Angles April 7, 2025 4 / 16 How to Find the Sine and Cosine of 45◦ = π 4 Special and Reference Angles April 7, 2025 5 / 16 How to Find the Sine and Cosine of 45◦ = π 4 Need to find the side lengths of a right triangle of hypotenuse 1 where one of the angles is 45◦ 90◦ + 45◦ + 45◦ = 180◦ , so both non-right angles are the same: 45◦ So the two legs opposite the non-right angles have the same length; call that length ℓ q √ By Pythagorean theorem, ℓ2 + ℓ2 = 1; solving for ℓ, we get 12 = 22 √ √ ◦ 2 ◦ 2 So, cos(45 ) = and sin(45 ) = 2 2 Special and Reference Angles April 7, 2025 5 / 16 How to Find the Sine and Cosine of 30◦ = π 6 Special and Reference Angles April 7, 2025 6 / 16 How to Find the Sine and Cosine of 30◦ = π 6 Need to find the side lengths of a right triangle of hypotenuse 1 where one of the angles is 30◦ Flip the triangle around its longer (adjacent to 30◦ ) side ℓ to make a bigger triangle That triangle’s angles are 180◦ − 90◦ − 30◦ = 60◦ , 60◦ , and 2(30◦ )60◦ So the triangle’s side lengths are all the same √ So 2h = 1 and h = 12 ; by Pythagorean theorem, ℓ = 23 √ ◦ 3 1 So, cos(30 ) = and sin(30◦ ) = 2 2 Special and Reference Angles April 7, 2025 6 / 16 How to Find the Sine and Cosine of 60◦ = π 3 Special and Reference Angles April 7, 2025 7 / 16 How to Find the Sine and Cosine of 60◦ = π 3 Need to find the side lengths of a right triangle of hypotenuse 1 where one of the angles is 60◦ We literally just did this! The shorter side (adjacent to 60◦ ) is 21 ; the √ longer side (opposite to 60◦ ) is 3 2 ; √ ◦ 1 ◦ 3 So, cos(60 ) = and sin(60 ) = 2 2 Special and Reference Angles April 7, 2025 7 / 16 With similar tricks, you can compute the exact values of sin and cos for even more angles! √ √ √ √ ◦ 6− 2 6+ 2 ◦ sin(15 ) = cos(15 ) = 4 4 √ p √ 5−1 10 + 2 5 sin(18◦ ) = ◦ cos(18 ) = 4 4... But for now, only worry about 0◦ , 30◦ , 45◦ , 60◦ , and 90◦ ! Special and Reference Angles April 7, 2025 8 / 16 With similar tricks, you can compute the exact values of sin and cos for even more angles! √ √ √ √ ◦ 6− 2 6+ 2 ◦ sin(15 ) = cos(15 ) = 4 4 √ p √ 5−1 10 + 2 5 sin(18◦ ) = ◦ cos(18 ) = 4 4... But for now, only worry about 0◦ , 30◦ , 45◦ , 60◦ , and 90◦ ! (Though I may have you try out some of these calculations for the Challenge Problem this week!) Special and Reference Angles April 7, 2025 8 / 16 Reference Angles Definition Let θ be an angle in the unit circle. The reference triangle of that angle is the right triangle with vertices (0, 0), (cos(θ), 0), and (cos(θ), sin(θ)). In other words, the reference triangle of θ is the right triangle “aligned with” the x-axis that contains both the origin and θ’s reference point. The reference angle of that angle is the interior angle of the reference triangle that is formed at the vertex (0, 0). Special and Reference Angles April 7, 2025 9 / 16 Escaping the First Quadrant Let α be the reference angle of an angle θ. α is equal to “how far away” θ is from either 0◦ = 0 or 180◦ = π. α will always be an angle in the first quadrant. | cos(θ)| = cos(α) and | sin(θ)| = sin(α). To get the signs of cos(θ) and sin(θ), check θ’s quadrant. Special and Reference Angles April 7, 2025 10 / 16 Example of Reference Angles What is cos(210◦ )? What is sin(210◦ )? Our reference angle is 30◦ , since 180◦ + 30◦ = 210◦ √ So, | cos(210◦ )| = 2 3 and | sin(210◦ )| = 1 2 We’re in Quadrant III, so both sin and cos will be negative √ So, cos(210◦ ) = − 2 3 and sin(210◦ ) = − 21 Special and Reference Angles April 7, 2025 11 / 16 In sum... The sin and cos of angles between 0◦ and 90◦ are intimately linked with right triangles of hypotenuse 1 The sin and cos of an angle anywhere else on the unit circle can be expressed in terms of the sin or cos between 0◦ and 90◦ (namely, its reference angle) Special and Reference Angles April 7, 2025 12 / 16 In sum... The sin and cos of angles between 0◦ and 90◦ are intimately linked with right triangles of hypotenuse 1 The sin and cos of an angle anywhere else on the unit circle can be expressed in terms of the sin or cos between 0◦ and 90◦ (namely, its reference angle) Let’s use these notions to fill our the rest of our unit circle chart! Special and Reference Angles April 7, 2025 12 / 16 “Reference Angles” on the Vertical Axis Sometimes, it’s also useful to draw triangles along the y -axis – i.e., with coordinates (0, 0), (0, sin(θ)), and (cos(θ), sin(θ)) Let β be the interior angle of that triangle along the vertex. Then: | cos(θ)| = sin(β) | sin(θ)| = cos(β) Theorem Let θ be an angle. Then: sin(θ) = cos(90◦ − θ) cos(θ) = sin(90◦ − θ) Special and Reference Angles April 7, 2025 13 / 16 Angles Outside of [0, 2π] We’ve mostly limited ourselves to angles between 0◦ = 0 and 360◦ = 2π so far – but we can extend our notions to angles beyond those numbers! What’s an angle above 2π, say, 2πn + α? Well, you do n full rotations counterclockwise, and then rotate α degrees counterclockwise! What’s a negative angle, say, −α? Well, it’s a rotation by α, but clockwise, in the opposite direction! Special and Reference Angles April 7, 2025 14 / 16 Sine and Cosine of Angles Outside of [0, 2π] As far as our reference points are concerned, rotating 360◦ = 2π (any number of times) is the same as not having rotated at all. So: Theorem Let θ be any angle. Then: sin(θ) = sin(θ + 2π) cos(θ) = cos(θ + 2π) To put it another way, the sin and cos functions are periodic, with a period of 360◦ = 2π. If you add or subtract 360◦ = 2π from an angle (any number of times), the angle’s sine and cosine stay the same. (We’ll learn more about periodic functions next week!) Special and Reference Angles April 7, 2025 15 / 16 Sine and Cosine of Angles Outside of [0, 2π], cont. Corollary Let θ be any angle. Then, there exists some angle θ0 ∈ [0, 2π) and some integer n ∈ {... , −2, −1, 0, 1, 2,... } such that θ = θ0 + 2πn. And furthermore: sin(θ) = sin(θ0 + 2πn) = sin(θ0 ) cos(θ) = cos(θ0 + 2πn) = cos(θ0 ) (Obviously, these formulas also work if you replace 2π with 360◦.) Examples: sin(765◦ ) sin(− π4 ) Special and Reference Angles April 7, 2025 16 / 16 Sine and Cosine of Angles Outside of [0, 2π], cont. Corollary Let θ be any angle. Then, there exists some angle θ0 ∈ [0, 2π) and some integer n ∈ {... , −2, −1, 0, 1, 2,... } such that θ = θ0 + 2πn. And furthermore: sin(θ) = sin(θ0 + 2πn) = sin(θ0 ) cos(θ) = cos(θ0 + 2πn) = cos(θ0 ) (Obviously, these formulas also work if you replace 2π with 360◦.) Examples: √ sin(765◦ ) = sin(2 · 360◦ + 45◦ ) = sin(45◦ ) = 2 2 sin(− π4 ) Special and Reference Angles April 7, 2025 16 / 16 Sine and Cosine of Angles Outside of [0, 2π], cont. Corollary Let θ be any angle. Then, there exists some angle θ0 ∈ [0, 2π) and some integer n ∈ {... , −2, −1, 0, 1, 2,... } such that θ = θ0 + 2πn. And furthermore: sin(θ) = sin(θ0 + 2πn) = sin(θ0 ) cos(θ) = cos(θ0 + 2πn) = cos(θ0 ) (Obviously, these formulas also work if you replace 2π with 360◦.) Examples: √ sin(765◦ ) = sin(2 · 360◦ + 45◦ ) = sin(45◦ ) = 2 2 √ sin(− π4 ) = sin(−1 · 2π + 7π 4 ) = sin( 7π 4 )= − 22 Special and Reference Angles April 7, 2025 16 / 16