What is the relation among all the trigonometrical ratios of (90° - θ)?
In trigonometrical ratios of angles (90° - θ) we will find the relation between all six trigonometrical ratios.
Let a rotating line OA rotates about O in the anti-clockwise direction, from initial position to ending position makes an angle ∠XOA = θ. Now a point C is taken on OA and draw CD perpendicular to OX or OX'.
Again another rotating line OB rotates about O in the anti-clockwise direction, from initial position to ending position (OX) makes an angle ∠XOY = 90°; this rotating line now rotates in the clockwise direction, starting from the position (OY) makes an angle ∠YOB = θ.
Now, we can observe that ∠XOB = 90° - θ.
Again a point E is taken on OB such that OC = OE and draw EF
perpendicular
to
OX or OX'.
Since, ∠YOB = ∠XOA
Therefore, ∠OEF = ∠COD.
Now, from the right-angled ∆EOF and right-angled ∆COD we get, ∠OEF = ∠COD and OE = OC.
Hence,
∆EOF ≅ ∆COD (congruent).
Therefore, FE = OD, OF = DC and OE = OC.
According to the definition of trigonometric ratio we get,
sin (90° - θ) = \(\frac{FE}{OE}\)
sin (90° - θ) = \(\frac{OD}{OC}\), [FE = OD and OE = OC, since ∆EOF ≅ ∆COD]
sin (90° - θ) = cos θ
cos (90° - θ) = \(\frac{OF}{OE}\)
cos (90° - θ) = \(\frac{DC}{OC}\), [OF = DC and OE = OC, since ∆EOF ≅ ∆COD]
cos (90° - θ) = sin θ
tan (90° - θ) = \(\frac{FE}{OF}\)
tan (90° - θ) = \(\frac{OD}{DC}\), [FE = OD and OF = DC, since ∆EOF ≅ ∆COD]
tan (90° - θ) = cot θ
Similarly, csc (90° - θ) = \(\frac{1}{sin (90° - \Theta)}\)
csc (90° - θ) = \(\frac{1}{cos \Theta}\)
csc (90° - θ) = sec θ
sec ( 90° - θ) = \(\frac{1}{cos (90° - \Theta)}\)
sec (90° - θ) = \(\frac{1}{sin \Theta}\)
sec (90° - θ) = csc θ
and cot (90° - θ) = \(\frac{1}{tan (90° - \Theta)}\)
cot (90° - θ) = \(\frac{1}{cot \Theta}\)
cot (90° - θ) = tan θ
Solved examples:
1. Find the value of cos 30°.
Solution:
cos 30° = sin (90 - 60)°
= sin 60°; since we know, cos (90° - θ) = sin θ
= \(\frac{√3}{2}\)
2. Find the value of csc 90°.
Solution:
csc 90° = csc (90 - 0)°
= sec 0°; since we know, csc (90° - θ) = sec θ
= 1
● Trigonometric Functions
11 and 12 Grade Math
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