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Signs of trigonometric ratios

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This presentation illustrate the use of signs of trigonometric ratios

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Signs of trigonometric ratios

  1. 1. Signs of trigonometric ratios by SBR www.harekrishnahub.com
  2. 2. www.harekrishnahub.com Consider a circle with centre 𝑢 and radius 𝒓 units. Let the circle cut the 𝒙 βˆ’ π’‚π’™π’Šπ’” at 𝑨 and 𝑨′ and π’š βˆ’ π’‚π’™π’Šπ’” at 𝑩 and 𝑩′. Let 𝑷 (𝒙, π’š) be any point on the circumference of the circle. Join 𝑢𝑷. Let the radius vector 𝑢𝑷 make an angle 𝜽 with the positive 𝒙 βˆ’ π’‚π’™π’Šπ’”. ∴ βˆ π‘Ώπ‘Άπ‘· = 𝜽 and 𝑢𝑷 = 𝒓 Draw 𝑷𝑴 βŸ‚ to the 𝒙 βˆ’ π’‚π’™π’Šπ’”. ∴ 𝑢𝑴 = 𝒙 and 𝑷𝑴 = π’š
  3. 3. www.harekrishnahub.com ∴ we have, Points Coordinates 𝑃 π‘₯, 𝑦 𝑀 π‘₯, 0 𝐴 π‘Ÿ, 0 𝐡 0, π‘Ÿ 𝐴′ βˆ’π‘Ÿ, 0 𝐡′ 0, βˆ’π‘Ÿ
  4. 4. www.harekrishnahub.com βŠΏπ‘·π‘΄π‘Ά form a right angle triangle. We have, π’”π’Šπ’ 𝜽 = π’š 𝒓 𝒄𝒐 𝒔 𝜽 = 𝒙 𝒓 𝒕𝒂𝒏 𝜽 = π’š 𝒙 𝒓 represents the magnitude (length) of the radius vector 𝑢𝑷. It is always positive. 𝑢𝑴 and 𝑷𝑴 represents the horizontal and vertical displacement components of point 𝑷. Hence, they can be positive or negative depending the quadrant in which the radius vector 𝑢𝑷 lies.
  5. 5. www.harekrishnahub.com Quadran t 𝛉 Sign Trigonometric ratios Remarks π‘₯ 𝑦 𝑠𝑖𝑛 πœƒ = 𝑦 π‘Ÿ π‘π‘œπ‘  πœƒ = π‘₯ π‘Ÿ π‘‘π‘Žπ‘› πœƒ = 𝑦 π‘₯ I 0 < πœƒ < 90 + + + + = + 𝑣𝑒 + + = + 𝑣𝑒 + + = + 𝑣𝑒 ALL +ve II 90 < πœƒ < 180 - + + + = + 𝑣𝑒 βˆ’ + = βˆ’ 𝑣𝑒 + βˆ’ = βˆ’ 𝑣𝑒 SIN +ve III 180 < πœƒ < 270 - - βˆ’ + = βˆ’ 𝑣𝑒 βˆ’ + = βˆ’ 𝑣𝑒 βˆ’ βˆ’ = + 𝑣𝑒 TAN +ve IV 270 < πœƒ < 360 + - βˆ’ + = βˆ’ 𝑣𝑒 + + = + 𝑣𝑒 βˆ’ + = βˆ’ 𝑣𝑒 COS +ve
  6. 6. www.harekrishnahub.com β€’ Thus, in the first quadrant, all of the trigonometric ratios are positive. β€’ In the second quadrant, only π’”π’Šπ’ 𝜽 and 𝒄𝒐𝒔𝒆𝒄 𝜽 are positive and all other ratios are negative. β€’ In the third quadrant, only tan 𝜽 and cot 𝜽 are positive and all other ratios are negative. β€’ In the fourth quadrant, only cos 𝜽 and sec 𝜽 are positive and all other ratios are negative.
  7. 7. www.harekrishnahub.com The following figure is useful in remembering the signs of the trigonometric ratios.
  8. 8. www.harekrishnahub.com When P Coincides with π‘₯ 𝑦 𝑠𝑖𝑛 πœƒ 𝑦 π‘Ÿ π‘π‘œπ‘  πœƒ π‘₯ π‘Ÿ π‘‘π‘Žπ‘› πœƒ 𝑦 π‘₯ 𝑨 (𝜽 = 𝟎) π‘Ÿ 0 𝑠𝑖𝑛 0 = 0 π‘π‘œπ‘  0 = 1 π‘‘π‘Žπ‘› 0 = 0 𝑩 𝜽 = πŸ—πŸŽ 0 π‘Ÿ 𝑠𝑖𝑛 90 = 1 π‘π‘œπ‘  90 = 0 π‘‘π‘Žπ‘› 90 = ∞ 𝑨′ 𝜽 = πŸπŸ–πŸŽ βˆ’π‘Ÿ 0 𝑠𝑖𝑛 180 = 0 π‘π‘œπ‘  180 = βˆ’1 π‘‘π‘Žπ‘› 180 = 0 𝑩′ 𝜽 = πŸπŸ•πŸŽ 0 βˆ’π‘Ÿ 𝑠𝑖𝑛 270 = βˆ’1 π‘π‘œπ‘  270 = 0 π‘‘π‘Žπ‘› 270 = βˆ’βˆž

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