# Trigonometrical Ratios of (- θ)

What is the relation among all the trigonometrical ratios of (– θ)?

In trigonometrical ratios of angles (- 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 OA make an angle ∠XOA = θ.

 Diagram 1 Diagram 2

Again a rotating line OA rotates about O in the clockwise direction and makes an angle ∠XOB having magnitude equal to ∠XOA.

Then we get, ∠XOB = - θ. Observe the diagram 1 and 4 to take a point C on OA and draw CD perpendicular to OX. Or we can also observe the diagram 2 and 3 where CD perpendicular to OX'. Let produce CD to intersect OB at E. Now, from the ∆ COD and ∆ EOD we get ∠COD = ∠EOD (same magnitude), ∠ODC = ∠ODE and OD is common.

Therefore, ∆ COD ≅ ∆ EOD (congruent)

Therefore,  according to the rules of trigonometric sign we get,

ED = - CD and OE = OC.

Again according to the definition of trigonometric ratios,

sin (- θ) = $$\frac{ED}{OE}$$

sin (- θ) = $$\frac{- CD}{OC}$$, [ED = CD and OE = OC since, ∆ COD ≅ ∆ EOD]

sin (- θ) = - sin θ

again, cos (- θ) = $$\frac{OD}{OE}$$

cos (- θ) = $$\frac{OD}{OC}$$, [OE = OC since, ∆ COD ≅ ∆ EOD]

cos (- θ) = cos θ

again, tan (- θ) = $$\frac{ED}{OD}$$

tan (- θ) = $$\frac{- CD}{OD}$$, [ED = CD since, ∆ COD ≅ ∆ EOD]

tan (- θ) = -  tan θ.

similarly, csc (- θ) = $$\frac{1}{sin (- \Theta)}$$

csc (- θ) = $$\frac{1}{- sin \Theta}$$

csc (- θ) = - csc θ.

again, sec (- θ) = $$\frac{1}{cos (- \Theta)}$$

sec (- θ) = $$\frac{1}{cos \Theta}$$

sec (- θ) = sec θ.

And again, cot (- θ) = $$\frac{1}{tan (- \Theta)}$$

cot (- θ) = $$\frac{1}{- tan \Theta}$$

cot (- θ) = - cot θ.

Solved example:

1. Find the value of sin (- 45)°.

Solution:

sin (- 45)° = - sin 45°; since we know sin (- θ) = - sin θ

= $$\frac{-1}{√2}$$

2. Find the value of sec (- 60)°.

Solution:

sec (- 60)° = sec 60°; since we know sec (- θ) = sec θ

= 2

3. Find the value of cot (- 90)°.

Solution:

cot (- 90)° = - tan 90°; since we know cot (- θ) = - tan θ

= 0

Trigonometric Functions

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