Trig Graphs And equations Revision A2

sin 𝜃 =− sin (−𝜃) sin 𝜃 = sin (180−𝜃) 𝑐𝑜𝑠 𝜃 = 𝑐𝑜s (−𝜃) Radians These are the Trigonometric graphs, but with radians instead… y y = sinθ sin 𝜃 =− sin (−𝜃) sin 𝜃 = sin (180−𝜃) cos 𝜃 = cos (360−𝜃) 𝑐𝑜𝑠 𝜃 = 𝑐𝑜s (−𝜃) tan 𝜃 =− tan (−𝜃) tan 𝜃 = tan (𝜃+180) Properties of the graphs Can you put them in words? 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 90º 180º π 3π 2 270º 360º π 2 2π -1 y y = cosθ 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 90º π 2 180º π 3π 2 270º 360º 2π -1 y = tanθ 1 θ -360º -2π -270º -3π 2 -180º -π -90º -π 2 π 2 90º 180º π 270º 3π 2 360º 2π -1

Maxima/Minima at (90,1) and (270,-1) (and every 180 from then) 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃 Reciprocal graphs These are the Reciprocal Trigonometric graphs 1 y = Sinθ 90 180 270 360 -1 Maxima/Minima at (90,1) and (270,-1) (and every 180 from then) 1 90 180 270 360 -1 Asymptotes at 0, 180, 360 (and every 180° from then) y = Cosecθ

Maxima/Minima at (0,1) (180,-1) and (360,1) (and every 180 from then) 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃 Reciprocal graphs These are the Reciprocal Trigonometric graphs 1 y = Cosθ 90 180 270 360 -1 Maxima/Minima at (0,1) (180,-1) and (360,1) (and every 180 from then) 1 90 180 270 360 -1 Asymptotes at 90 and 270 (and every 180° from then) y = Secθ

𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃 Reciprocal graphs These are the Reciprocal Trigonometric graphs y = Tanθ 90 180 270 360 Asymptotes at 0, 180 and 360 (and every 180° from then) 90 180 270 360 y = Cotθ

𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 Trig Identities Can you write ten useful Trig identities 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃 𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃 𝑎 2 = 𝑏 2 + 𝑐 2 −2𝑏𝑐 cos 𝐴 sin 𝐴 𝑎 = sin 𝐵 𝑏 𝑠𝑖𝑛 2 𝑥=1− 𝑐𝑜𝑠 2 𝑥 𝑐𝑜𝑠 2 𝑥=1− 𝑠𝑖𝑛 2 𝑥

Trig Identities Can you write ten useful Trig identities Formula booklet

A. Solve, for 0 ≤ x < 2π, 2 cot (𝑥− 𝜋 3 ) =5⁡   Give your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)

A. Solve, for 0 ≤ x < 2π, 2 cot (𝑥− 𝜋 3 ) =5⁡   Give your answers to two decimal places (Solutions based entirely on graphical or numerical methods are not acceptable.) (4) SOLUTION and MARKSCHEME cot 𝑥− 𝜋 3 = 5 2 tan 𝑥− 𝜋 3 = 2 5 𝐵1 𝑥− 𝜋 3 =0.381 gives 𝑥=1.43 𝑀1 𝐴1 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑎𝑛𝑠𝑤𝑒𝑟 𝑥− 𝜋 3 =0.381, 3.52,, … gives 𝑥=4.57 𝐴1 𝑓𝑜𝑟 2𝑛𝑑 𝑎𝑛𝑠𝑤𝑒𝑟

Give your answers to two decimal places B. Solve, for 0<𝑥< 3𝜋 2 , 𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐 2 𝑥=4   Give your answers to two decimal places (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)

Give your answers to two decimal places B. Solve, for 0<𝑥< 3𝜋 2 , 𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐 2 𝑥=4   Give your answers to two decimal places (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) SOLUTION and MARKSCHEME 𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 1+ 𝑐𝑜𝑡 2 𝑥 =4 𝐵1 𝑢𝑠𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑡𝑜 𝑚𝑎𝑘𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 2 𝑐𝑜𝑡 2 𝑥 − cot 𝑥 − 3 = 0 2 cot 𝑥 −3 cot 𝑥 +1 = 0 𝑀1 𝑠𝑜𝑙𝑣𝑖𝑛𝑔 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝐴1 𝐴1 𝑒𝑎𝑐ℎ 𝑎𝑛𝑠𝑤𝑒𝑟 cot 𝑥 = 3 2 rearrange to t𝑎𝑛 𝑥 = 2 3 gives 𝑥=0.588, 3.73 cot 𝑥 =−1 rearrange to t𝑎𝑛 𝑥 =−1 gives 𝑥= 3𝜋 4 𝑜𝑛𝑙𝑦 𝐴1 𝑓𝑜𝑟 3𝑟𝑑 𝑎𝑛𝑠𝑤𝑒𝑟

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