EXACT Trig Ratios in radians (full lesson) | grade 12 MHF4U

EXACT Trig Ratios in radians (full lesson) | grade 12 MHF4U | jensenmath.ca

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@Dan-fo2rf 2 жыл бұрын

This was the best way to review grade 11 and key concepts of grade 12 in the most efficient amount of time possible. It’s insane how easy u made this look , and how your able to explain key concepts so well. Your an amazing teacher!

@geoff-n4e 11 ай бұрын

🎯 Key Takeaways for quick navigation: [00:18] *Key trig concepts review: special triangles, unit circle, CAST rule, reference angles, SohCahToa.* [00:46] *SohCahToa: acronym for ratios of sides in a right triangle based on sine, cosine, and tangent.* [01:29] *Similar triangles with the same reference angle have identical sine & tangent ratios.* [01:56] *Unit circle demonstration: calculating sine, cosine, and tangent for a 40° reference angle.* [02:24] *Changing the reference angle changes the trig ratios for the original triangle, but not for similar triangles with the same new angle.* [05:16] 🟥🇧 *Key takeaway: Red angle (reference) & blue angle (principal) visualized: reference angle from terminal arm to closest x-axis, principal angle from positive x-axis to terminal arm.* [05:29] ⭕ *Key takeaway: Unit circle is crucial for trigonometry understanding - radius of 1 allows relating x/y coordinates on circle to sine/cosine of angles.* [05:57] *Key takeaway: Finding coordinates on unit circle for a rotated angle reveals its sine & cosine: x = cos(θ), y = sin(θ).* [06:39] ➡️⭕ *Key takeaway: SOH CAH TOA in action: sin(θ) = y/1 (opposite over hypotenuse on unit circle), cos(θ) = x/1 (adjacent over hypotenuse).* [07:20] *Key takeaway: Unit circle coordinates & sine/cosine relationship are fundamental for understanding CAST rule and future trig identities.* [07:59] *CAST rule overview: order (CAS-T) tells which trig ratios are positive in each quadrant.* [08:13] Quadrant *signs: Sine positive in 1 & 2, Cosine positive in 1 & 4, Tangent positive in 1 & 3 (based on CAST).* [08:40] Sine *and y-coordinate connection: sine positive when y > 0 (above x-axis), negative when y < 0 (below x-axis).* [09:08] Cosine *and x-coordinate connection: cosine positive when x > 0, negative when x < 0.* [09:23] Tangent *and quadrant connections: tangent positive when both x & y positive (quadrant 1) or both negative (quadrant 3).* [09:52] CAST *rule reminder: CAST helps determine quadrant positivity for sine, cosine, and tangent based on their x and y relationships in the unit circle.* [10:07] Special *triangle: 45-45-90 triangle (isosceles, 45° angles in radians = pi/4, side lengths 1:1:√2).* [13:24] *CAST rule and quadrant: Cosine quadrant (CAST) means only cosine is positive, use tan of reference angle with negative sign for other ratios.* [13:51] *Reference angle & tan: tan(11π/6) = -tan(π/6) due to quadrant and CAST rule.* [14:06] *Special triangle & tan: tan(π/6) = 1/√3 from 30-60-90 triangle, so tan(11π/6) = -√3/3.* [14:49] *Quadrant and angle rewrite: 5π/6 in quadrant 2, rewrite angles with common denominator for easier visualization.* [15:20] *Terminal arm and angle location: 5π/6 between 3π/6 and 5π/6 on unit circle.* [15:34] *Reference angle for sine: β = 6π/6 - 5π/6 = π/6 for angle of 5π/6.* [16:01] *CAST rule and quadrant: Sine quadrant (CAST) means sine is positive for 5π/6.* [16:13] CAST *rule and quadrant: Sine quadrant (CAST) means sine is positive, so sine of 5π/6 can be found using sine of its reference angle (π/6) without modifying the sign.* [16:42] Special *triangle and sine: Sine of π/6 = 1/2 from 30-60-90 triangle, so sine of 5π/6 = 1/2 (same y-coordinate on unit circle).* [16:56] Reference *angle equivalence: Terminal arms for π/6 and 5π/6 share the same y-coordinate on the unit circle due to overlapping rotations, justifying using sine of π/6 for 5π/6.* [18:10] Cosine *angle location: 5π/4 in quadrant 3, between 4π/4 and 6π/4 on the unit circle.* [18:40] CAST *rule and quadrant: Quadrant 3 (CAST) means cosine and tangent are negative, so we need to find the reference angle for cosine of 5π/4.* [19:10] Reference *angle for cosine in quadrant 3: β = π/4 for cosine of 5π/4 due to rotation from terminal arm to closest x-axis.* [19:24] CAST *rule and quadrant: Quadrant 3 (CAST) means cosine is negative, so cosine of 5π/4 will be found using negative cosine of its reference angle (π/4).* [19:52] Special *triangle and cosine: Cosine of π/4 = 1/√2 from 30-60-90 triangle, so cosine of 5π/4 = -√2/2 (negative due to CAST rule).* [20:06] Rationalizing *denominator: -√2/2 simplifies to -2 after multiplying top and bottom by √2.* [20:21] Secant *as reciprocal of cosine: sec(x) = 1 / cos(x) identity used to evaluate secant of 3π/4.* [20:49] Cosine *angle location: 3π/4 in quadrant 2, between 2π/4 and 4π/4 on the unit circle.* [21:16] Reference *angle for cosine in quadrant 2: β = π - 3π/4 = π/4 due to rotation from terminal arm to closest x-axis.* [21:30] CAST *rule and quadrant: Quadrant 2 (CAST) means cosine is negative, so cosine of 3π/4 will be found using negative cosine of its reference angle (π/4).* [21:44] Special *triangle and cosine: Cosine of π/4 = 1/√2 from 30-60-90 triangle.* [21:57] Secant *calculation: sec(3π/4) = 1 / (-cos(π/4)) = 1 / (-1/√2) = -√2.* [22:12] Identifying *quadrant and angle: 5π/3 in quadrant 4, rewritable with denominator 3 as 1.5π/3, 3π/3, 4.5π/3, and 6π/3.* [22:38] Rotating *terminal arm and labeling: 5π/3 rotation from positive x-axis, tip labeled as principal angle.* [22:53] Reference *angle calculation: β = 6π/3 - 5π/3 = 1π/3 for angle between terminal arm and closest x-axis.* [23:07] CAST *rule and quadrant: Quadrant 4 (CAST) means cosine positive, sine and tangent negative.* [23:35] Evaluating *sine with reference angle: sin(5π/3) = -sin(π/3) due to CAST rule and same absolute y-coordinate on unit circle despite different intersections.* [24:03] Special *triangle and sine: sin(π/3) = √3/2 from 30-60-90 triangle, so sin(5π/3) = -√3/2 (negative due to CAST rule).* [25:00] Sine *calculation on unit circle: sin(5π/3) = -√3/2 and sin(π/3) = √3/2 due to opposite y-coordinates with different signs.* [25:14] Cosine *in quadrant 4 (CAST rule): cos(5π/3) = cos(π/3) = 1/2 as cosine is positive in quadrant 4.* [25:45] Tangent *in quadrant 4 (CAST rule): tan(5π/3) = -tan(π/3) = -√3 as tangent is negative in quadrant 4.* [26:13] Cosecant *calculation: cosec(5π/3) = 1 / sin(5π/3) = -2/√3 (reciprocal of sine).* [26:41] Secant *calculation: sec(5π/3) = 1 / cos(5π/3) = 2 (reciprocal of cosine).* [26:54] Cotangent *calculation: cot(5π/3) = 1 / tan(5π/3) = -√3 / tan(5π/3) = -1/√3 (reciprocal of tangent).* [27:10] Real-world *application: Justin's kite flies higher, increasing the angle from π/6 to π/3 with the ground. We need to find the horizontal distance gained due to the kite's new position.* [27:40] Identifying *problem: Need to find horizontal distance gained by kite after flying higher, represented by difference in x-coordinates of initial and final positions.* [28:06] Formulating *calculation: Horizontal distance equals x-coordinate 1 minus x-coordinate 2 (length of yellow side).* [28:20] Triangle *analysis: Dividing problem into two right triangles (blue and red) to find x-coordinates (x1 and x2) of kite positions.* [28:46] Redrawing *triangles: Blue triangle with angle π/6, hypotenuse 50 m, and unknown side x1; red triangle with angle π/3, hypotenuse 50 m, and unknown side x2.* [29:16] Determining *x2: Using cosine and right triangle properties, 50 cos(π/3) = x2, with cos(π/3) = 1/2 from special triangle, resulting in x2 = 50 * 1/2 = 25 meters.* [29:43] Determining *x1: Using cosine and right triangle properties, 50 cos(π/6) = x1, with cos(π/6) = √3/2 from special triangle, resulting in x1 = 50 * √3/2 = 25√3 meters.* [30:26] Final *answer: Horizontal distance gained by kite is x1 - x2 = 25√3 meters - 25 meters = 25(√3 - 1) meters (exact expression as specified in the question).* Made with HARPA AI

@ociones 10 ай бұрын

THANKS oh u used ai