This presentation illustrate the use of signs of trigonometric ratios

1. Signs of trigonometric ratios
by
SBR
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Consider a circle with centre 𝑶 and radius 𝒓 units.
Let the circle cut the 𝒙 − 𝒂𝒙𝒊𝒔 at 𝑨 and 𝑨′ and 𝒚 − 𝒂𝒙𝒊𝒔 at 𝑩 and 𝑩′.
Let 𝑷 (𝒙, 𝒚) be any point on the circumference of the circle.
Join 𝑶𝑷.
Let the radius vector 𝑶𝑷 make an angle 𝜽 with the positive 𝒙 − 𝒂𝒙𝒊𝒔.
∴ ∠𝑿𝑶𝑷 = 𝜽 and 𝑶𝑷 = 𝒓
Draw 𝑷𝑴 ⟂ to the 𝒙 − 𝒂𝒙𝒊𝒔.
∴ 𝑶𝑴 = 𝒙 and 𝑷𝑴 = 𝒚

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∴ we have,
Points Coordinates
𝑃 𝑥, 𝑦
𝑀 𝑥, 0
𝐴 𝑟, 0
𝐵 0, 𝑟
𝐴′ −𝑟, 0
𝐵′ 0, −𝑟

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⊿𝑷𝑴𝑶 form a right angle triangle.
We have,
𝒔𝒊𝒏 𝜽 =
𝒚
𝒓 𝒄𝒐 𝒔 𝜽 =
𝒙
𝒓
𝒕𝒂𝒏 𝜽 =
𝒚
𝒙
𝒓 represents the magnitude (length) of the radius vector 𝑶𝑷. It is always
positive.
𝑶𝑴 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 𝑶𝑷 lies.

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Quadran
t
𝛉
Sign Trigonometric ratios
Remarks
𝑥 𝑦 𝑠𝑖𝑛 𝜃 =
𝑦
𝑟
𝑐𝑜𝑠 𝜃 =
𝑥
𝑟
𝑡𝑎𝑛 𝜃 =
𝑦
𝑥
I 0 < 𝜃 < 90 + + +
+
= + 𝑣𝑒
+
+
= + 𝑣𝑒
+
+
= + 𝑣𝑒 ALL +ve
II 90 < 𝜃 < 180 - + +
+
= + 𝑣𝑒
−
+
= − 𝑣𝑒
+
−
= − 𝑣𝑒 SIN +ve
III 180 < 𝜃 < 270 - - −
+
= − 𝑣𝑒
−
+
= − 𝑣𝑒
−
−
= + 𝑣𝑒 TAN +ve
IV 270 < 𝜃 < 360 + - −
+
= − 𝑣𝑒
+
+
= + 𝑣𝑒
−
+
= − 𝑣𝑒 COS +ve

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• Thus, in the first quadrant, all of the trigonometric ratios are positive.
• In the second quadrant, only 𝒔𝒊𝒏 𝜽 and 𝒄𝒐𝒔𝒆𝒄 𝜽 are positive and all other
ratios are negative.
• In the third quadrant, only tan 𝜽 and cot 𝜽 are positive and all other ratios are
negative.
• In the fourth quadrant, only cos 𝜽 and sec 𝜽 are positive and all other ratios
are negative.

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The following figure is useful in remembering the signs of the trigonometric
ratios.

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When P
Coincides
with
𝑥 𝑦
𝑠𝑖𝑛 𝜃
𝑦
𝑟
𝑐𝑜𝑠 𝜃
𝑥
𝑟
𝑡𝑎𝑛 𝜃
𝑦
𝑥
𝑨
(𝜽 = 𝟎)
𝑟 0 𝑠𝑖𝑛 0 = 0 𝑐𝑜𝑠 0 = 1 𝑡𝑎𝑛 0 = 0
𝑩
𝜽 = 𝟗𝟎
0 𝑟 𝑠𝑖𝑛 90 = 1 𝑐𝑜𝑠 90 = 0 𝑡𝑎𝑛 90 = ∞
𝑨′
𝜽 = 𝟏𝟖𝟎
−𝑟 0 𝑠𝑖𝑛 180 = 0 𝑐𝑜𝑠 180 = −1 𝑡𝑎𝑛 180 = 0
𝑩′
𝜽 = 𝟐𝟕𝟎
0 −𝑟 𝑠𝑖𝑛 270 = −1 𝑐𝑜𝑠 270 = 0 𝑡𝑎𝑛 270 = −∞