The Tangent Function

The Tangent Function April 14, 2025 Tangent April 14, 2025 1 / 11 Agenda for the Day Learn what the tangent function is Learn several applications of the tangent function Tangent April 14, 2025 2 / 11 Logistical Note...

The Tangent Function April 14, 2025 Tangent April 14, 2025 1 / 11 Agenda for the Day Learn what the tangent function is Learn several applications of the tangent function Tangent April 14, 2025 2 / 11 Logistical Notes I’ve graded Challenge Problem 2; I’ll hand it back now! People did better on this one! Maybe I did a better job of guiding people Or maybe you’re all just getting better at trigonometry! Though alas, a lot of people didn’t get to the part about finding sines, which was the bit that actually tied in with the week’s lesson On a lot of people’s packets that I√handed back, I wrote something √ √ 1+ 5 10−2 5 like, “Letting A = 4 and B = 4...” This isn’t standard notation √ or a√ standard thing to do or anything – I 10−2 5 just didn’t want to write 4 a hundred times Due to the way the course topics shook out, we’re doing this week’s Challenge Problem today, and the double-lecture on Wednesday! Remember WeBWorKs and Canvas Quizzes! Make use of office hours and “Email Instructor”! Remember that there’s a midterm next week Wednesday (April 24) I’ll pass out study guides Wednesday this week Tangent April 14, 2025 3 / 11 What is the tangent function? Definition Let θ be an angle such that cos(θ) ̸= 0. Then the tangent of θ is given by: sin(θ) tan(θ) = cos(θ) For example: tan(30◦ ) = Tangent April 14, 2025 4 / 11 What is the tangent function? Definition Let θ be an angle such that cos(θ) ̸= 0. Then the tangent of θ is given by: sin(θ) tan(θ) = cos(θ) For example: 1 √ ◦ sin(30◦ ) 1 3 tan(30 ) = = √2 =√ = cos(30◦ ) 3 3 3 2 Tangent April 14, 2025 4 / 11 What is the tangent function? Definition Let θ be an angle such that cos(θ) ̸= 0. Then the tangent of θ is given by: sin(θ) tan(θ) = cos(θ) For example: 1 √ ◦ sin(30◦ ) 1 3 tan(30 ) = = √2 =√ = cos(30◦ ) 3 3 3 2 Let’s compute some more values of the tangent function, and fill out the rest of our unit circle sheet! Tangent April 14, 2025 4 / 11 Why do we care about this? Sure, we can divide sin(θ) by cos(θ) and get a new function. So what? We can divide any functions? What makes dividing them special? Why not add them or multiply them? Well, let’s see why we care about the tangent function! Tangent April 14, 2025 5 / 11 Application 1: Triangle Sides (SOHCAHTOA) O We already know that sin(θ) = H A and cos(θ) = H. What about tan(θ)? H O θ sin(θ) tan(θ) = A cos(θ) Tangent April 14, 2025 6 / 11 Application 1: Triangle Sides (SOHCAHTOA) O We already know that sin(θ) = H A and cos(θ) = H. What about tan(θ)? H O θ sin(θ) O/H O tan(θ) = = = A cos(θ) A/H A Theorem Let θ be a non-right angle in a right triangle. Let H be the length of the right triangle’s hypotenuse, let O be the length of the right triangle’s leg opposite to θ, and let A be the length of the right triangle’s leg adjacent to θ. Then: O A O sin(θ) = cos(θ) = tan(θ) = H H A Tangent April 14, 2025 6 / 11 Example of this Application Find ℓ. Tangent April 14, 2025 7 / 11 Example of this Application Find ℓ. Method 1 (if we don’t know about tan): 12 12 cos(19◦ ) = =⇒ h = ≈ 12.6914 h cos(19◦ ) ℓ sin(19◦ ) = =⇒ ℓ = 12.6914 sin(19◦ ) ≈ 4.1319 12.6914 Method 2 (if we don’t know about tan): 12 12 cos(19◦ ) = =⇒ h = ≈ 12.6914 h cos(19◦ ) p ℓ2 + 122 = 12.69142 =⇒ ℓ = 12.69142 − 122 ≈ 4.1319 Tangent April 14, 2025 7 / 11 Example of this Application Find ℓ. Method 3 (if we DO know about tan): ℓ tan(19◦ ) = =⇒ ℓ = 12 tan(19◦ ) ≈ 4.1319 12 Way more efficient! We don’t have to do the annoying intermediate step of finding the hypotenuse! Tangent April 14, 2025 7 / 11 Application 2: Slopes of Lines 10 y Say a line makes an angle of θ with the horizontal. What is its slope? 5 rise θ x slope −2 2run 4 6 8 10 Tangent April 14, 2025 8 / 11 Application 2: Slopes of Lines 10 y Say a line makes an angle of θ with the horizontal. What is its slope? 5 rise rise θ x slope = run −2 2run 4 6 8 10 Tangent April 14, 2025 8 / 11 Application 2: Slopes of Lines Say a line makes an angle of θ with 10 y the horizontal. What is its slope? 5 rise rise slope = run θ length of opposite side to θ x = −2 2run 4 6 8 10 length of adjacent side to θ = tan(θ) Tangent April 14, 2025 8 / 11 Application 2: Slopes of Lines Say a line makes an angle of θ with 10 y the horizontal. What is its slope? 5 rise rise slope = run θ length of opposite side to θ x = −2 2run 4 6 8 10 length of adjacent side to θ = tan(θ) Theorem Suppose a line y = mx + b in the coordinate plane makes an angle of θ with the positive x-axis. Then its slope is m = tan(θ) Tangent April 14, 2025 8 / 11 Broad Intuition for the Three Big Trigonometric Functions sine means height cosine means length tangent means slope (You know, usually. When you’re working with the coordinate plane like this.) Tangent April 14, 2025 9 / 11 More Intuition for the Three Big Trigonometric Functions The sin, cos, and tan functions map from angles to numbers/ratios. Your input for these functions should always be an angle! sin,cos,tan angles numbers/ratios The sin or cos of an angle will always be in the interval [−1, 1]. The tan of an angle can potentially by any real number. Tangent April 14, 2025 10 / 11 Example of this Application What is the equation for this line? (That is, what are m and b in this line’s y = mx + b?) Tangent April 14, 2025 11 / 11 Example of this Application What is the equation for this line? (That is, what are m and b in this line’s y = mx + b?) Step 1: Find m. m = tan(42◦ ) ≈ 0.9 Tangent April 14, 2025 11 / 11 Example of this Application What is the equation for this line? (That is, what are m and b in this line’s y = mx + b?) Step 1: Find m. m = tan(42◦ ) ≈ 0.9 Step 2: Find b. (We’ll use the point (−2, 0) on the line!) y = mx + b =⇒ 0 = 0.9(−2) + b =⇒ 0 = −1.8 + b =⇒ b = 1.8 Tangent April 14, 2025 11 / 11 Example of this Application What is the equation for this line? (That is, what are m and b in this line’s y = mx + b?) Step 1: Find m. m = tan(42◦ ) ≈ 0.9 Step 2: Find b. (We’ll use the point (−2, 0) on the line!) y = mx + b =⇒ 0 = 0.9(−2) + b =⇒ 0 = −1.8 + b =⇒ b = 1.8 So, y = 0.9x + 1.8 Tangent April 14, 2025 11 / 11