Trigonometric ratios relate the sides of a right triangle to an angle of the triangle. When a point P lies on a circle of radius r, the line segments from P to the center O and x-axis form a right triangle. The trigonometric ratios are defined as the ratio of an opposite, adjacent, or hypotenuse side to the hypotenuse. For example, sine is the ratio of the opposite side to the hypotenuse. Reciprocal trigonometric functions can also be defined, with sine and cosecant being reciprocals of each other.

1. Trigonometric Ratios
by
SBR
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Consider a circle with centre 𝑶 and radius 𝒓
units. 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 ⟂ the 𝒙 − 𝒂𝒙𝒊𝒔.
∴ 𝑶𝑴 = 𝒙 and 𝑷𝑴 = 𝒚

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𝑷𝑴𝑶 form a right angle triangle as shown
below.
Let us identify each of the sides of the triangle.
𝑶𝑷 = 𝒓 = 𝒙 𝟐 + 𝒚 𝟐 is the hypotenuse
𝑷𝑴 = 𝒚, the side opposite to 𝜽 is called the
opposite side.
𝑶𝑴 = 𝒙, the side adjacent to 𝜽 is called the
adjacent side.

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6 trigonometric ratios
Sine of angle θ 𝑠𝑖𝑛 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒚
𝒓
=
𝒚
𝒙 𝟐 + 𝒚 𝟐
CoSine of angle θ 𝑐𝑜𝑠 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝒙
𝒓
=
𝒙
𝒙 𝟐 + 𝒚 𝟐
Tangent of angle θ 𝑡𝑎𝑛 𝜃
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝒚
𝒙
for x ≠ 0
CoSecant of angle
θ
𝑐𝑜𝑠𝑒𝑐 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝒓
𝒚
=
𝒙 𝟐 + 𝒚 𝟐
𝒚
for y ≠ 0
Secant of angle θ 𝑠𝑒𝑐 𝜃
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝒓
𝒙
=
𝒙 𝟐 + 𝒚 𝟐
𝒙
for x ≠ 0
CoTangent of angle
θ
𝑐𝑜𝑡 𝜃
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝒙
𝒚
for y ≠ 0

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But
𝒕𝒂𝒏 𝜽 =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆
We have,
𝒕𝒂𝒏 𝜽 =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
Dividing both the numerator and the denominator by hypotenuse, we get
𝒄𝒐𝒔 𝜽 =
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒔𝒊𝒏 𝜽 =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
∴ 𝒕𝒂𝒏 𝜽 =
𝒔𝒊𝒏 𝜽
𝒄𝒐𝒔 𝜽

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Reciprocal relations
𝒔𝒊𝒏 𝜽 =
𝑶𝒑𝒑 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑
=
𝟏
𝒉𝒚𝒑
𝑶𝒑𝒑 𝒔𝒊𝒅𝒆
=
𝟏
𝒄𝒐𝒔𝒆𝒄 𝜽
𝒄𝒐𝒔 𝜽 =
𝒂𝒅𝒋 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑
=
𝟏
𝒂𝒅𝒋
𝑶𝒑𝒑 𝒔𝒊𝒅𝒆
=
𝟏
𝒔𝒆𝒄 𝜽
𝒕𝒂𝒏 𝜽 =
𝑶𝒑𝒑 𝒔𝒊𝒅𝒆
𝒂𝒅𝒋 𝒔𝒊𝒅𝒆
=
𝟏
𝒂𝒅𝒋 𝒔𝒊𝒅𝒆
𝑶𝒑𝒑 𝒔𝒊𝒅𝒆
=
𝟏
𝒄𝒐𝒕 𝜽
Therefore, 𝒔𝒊𝒏 𝜽 and 𝒄𝒐𝒔𝒆𝒄 𝜽 are
reciprocal to each other
Therefore, 𝒄𝒐𝒔 𝜽 and 𝒔𝒆𝒄 𝜽 are
reciprocal to each other
Therefore, 𝒕𝒂𝒏 𝜽 and 𝒄𝒐𝒕 𝜽 are
reciprocal to each other