Presentation on theme: "Chapter 6: Trigonometry 6"— Presentation transcript:
1 Chapter 6: Trigonometry 6 Chapter 6: Trigonometry 6.4: Trigonometric Functions (Wave Function Approach)Essential Question:What trigonometric function is represented by the x-coordinate and what trigonometric functions is represented by the y-coordinate?
2 6.4: Trigonometric Functions A note before we beginThis PowerPoint is going to deviate significantly from the book. You’re more than welcome to try and interpret the book’s unit circle approach, but the problems assigned can be solved using the wave approach in the following slides.We’ll discuss the unit circle approach Friday, and I want your honest opinion on what you feel is the easier way to understand trig functions.
3 6.4: Trigonometric Functions The sine curveThe sine (sin) curve is a cycle that starts at 0, reaches a max of 1 (at π/2), and a min of -1 (at 3π/2), and cycles every 2π
4 6.4: Trigonometric Functions The cosine curveThe cosine (cos) curve is a cycle that starts at 1, also reaches a max of 1 (at 0), a min of -1 (at π), and cycles every 2π
5 6.4: Trigonometric Functions The tangent curvetan θ = sin θ / cos θWhere cos θ = 0, the tangent is undefinedIt spikes up (like x3) in between those undefined pointsWhere sin θ = 0, tan θ = 0Unlike sin/cos, tan repeats every π radians
6 6.4: Trigonometric Functions Exact values of trigonometric functionsFind the exact values of the sine, cosine, and tangent functions when t = 0, π/2, π, 3π/2 and 2πSin functionZooming in on the sin graphThe sin functions starts at 0, so:sin 0 = 0sin π/2 = 1sin π = 0sin 3π/2 = -1sin 2π = 0Cos functionWorks the same way as the sin function, except cos starts at 1, and begins going downcos 0 = 1cos π/2 = 0cos π = -1cos 3π/2 = 0cos 2π = 1
7 6.4: Trigonometric Functions Exact values of trigonometric functionsFind the exact values of the sine, cosine, and tangent functions when t = 0, π/2, π, 3π/2 and 2πtan functionThe first sin/cos columns are from the last slide:tsin tcos ttan t = sin t/cos t1π/2undefinedπ-13π/22π
8 6.4: Trigonometric Functions Trigonometric Ratios of Coterminal AnglesFind the sine, cosine, and tangent of 7π/37π/3 is greater than 2π, so subtract to give us something to work with7π/3 – 2π = 7π/3 – 6π/3 = π/3Personally, I find it easier to think about the waves if you’re thinking in degreesπ/3 • 180°/π = 60° → which is one of our special anglesThe special angles count up… 0, 30, 45, 60, 90, …sin t starts at 0 and counts up…60 matches up withcos t starts at 1, and counts downtan t = sin t / cos t =
9 6.4: Trigonometric Functions Hints with angles greater than π (90°)As I mentioned back in section 6.1, the only values you’ll have to memorize are the sin and cosine values. All other trigonometric values can be found based off sin & cos (we’ll explore further in 6.5)This pattern continues until the cycle repeats.Angle0°30°45°60°90°120°135°150°180°210°cos θ1-1sin θ
10 6.4: Trigonometric Functions AssignmentPage 452Problems 11-23, (odd)For identifying the quadrants in use (cos x, sin x) to identify the quadrant.For example, let’s look at:#16) 11 [Note: we’re in radian mode in the calculators]cos 11 ≈sin 11 ≈We’re at the point (0.0044, ) which is in the 4th quadrantFor problems 31 – 53, remember your two numbers to memorize. Leave everything in fraction form (including 0.5 as ½)