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C HAPTER 6: T RIGONOMETRY 6.4: T RIGONOMETRIC F UNCTIONS (W AVE F UNCTION A PPROACH ) Essential Question: What trigonometric function is represented by the x-coordinate and what trigonometric functions is represented by the y-coordinate?

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6.4: T RIGONOMETRIC F UNCTIONS A note before we begin This 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.

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6.4: T RIGONOMETRIC F UNCTIONS The sine curve The sine (sin) curve is a cycle that starts at 0, reaches a max of 1 (at π /2), and a min of -1 (at 3 π /2), and cycles every 2 π

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6.4: T RIGONOMETRIC F UNCTIONS The cosine curve The 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 2 π

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6.4: T RIGONOMETRIC F UNCTIONS The tangent curve tan θ = sin θ / cos θ Where cos θ = 0, the tangent is undefined It spikes up (like x 3 ) in between those undefined points Where sin θ = 0, tan θ = 0 Unlike sin/cos, tan repeats every π radians

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6.4: T RIGONOMETRIC F UNCTIONS Exact values of trigonometric functions Find the exact values of the sine, cosine, and tangent functions when t = 0, π /2, π, 3 π /2 and 2 π Sin function Zooming in on the sin graph The sin functions starts at 0, so: sin 0 = 0 sin π /2 = 1 sin π = 0 sin 3 π /2 = -1 sin 2 π = 0 Cos function Works the same way as the sin function, except cos starts at 1, and begins going down cos 0 = 1 cos π /2 = 0 cos π = -1 cos 3 π /2 = 0 cos 2 π = 1

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6.4: T RIGONOMETRIC F UNCTIONS Exact values of trigonometric functions Find the exact values of the sine, cosine, and tangent functions when t = 0, π /2, π, 3 π /2 and 2 π tan function The first sin/cos columns are from the last slide: tsin tcos ttan t = sin t/cos t 0010 π /210undefined π 00 3 π /20undefined 2π2π 010

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6.4: T RIGONOMETRIC F UNCTIONS Trigonometric Ratios of Coterminal Angles Find the sine, cosine, and tangent of 7π/3 7 π /3 is greater than 2 π, so subtract to give us something to work with 7 π /3 – 2 π = 7 π /3 – 6 π /3 = π /3 Personally, I find it easier to think about the waves if you’re thinking in degrees π /3 180 ° / π = 60 ° → which is one of our special angles The special angles count up… 0, 30, 45, 60, 90, … sin t starts at 0 and counts up… 60 matches up with cos t starts at 1, and counts down 60 matches up with tan t = sin t / cos t =

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6.4: T RIGONOMETRIC F UNCTIONS 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. Angle 0°0° 30 ° 45 ° 60 ° 90 ° 120 ° 135 ° 150 ° 180 ° 210 ° cos θ 10 sin θ 010

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6.4: T RIGONOMETRIC F UNCTIONS Assignment Page 452 Problems 11-23, 31-53 (odd) For identifying the quadrants in 15-23 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 ≈ 0.0044 sin 11 ≈ -0.9999 We’re at the point (0.0044, -0.9999) which is in the 4 th quadrant For problems 31 – 53, remember your two numbers to memorize. Leave everything in fraction form (including 0.5 as ½)

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