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.

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**

**Basic Trigonometric Ratios and Their Names****Restrictions of Trigonometrical Ratios****Reciprocal Relations of Trigonometric Ratios****Quotient Relations of Trigonometric Ratios****Limit of Trigonometric Ratios****Trigonometrical Identity****Problems on Trigonometric Identities****Elimination of Trigonometric Ratios****Eliminate Theta between the equations****Problems on Eliminate Theta****Trig Ratio Problems****Proving Trigonometric Ratios****Trig Ratios Proving Problems****Verify Trigonometric Identities****Trigonometrical Ratios of 0°****Trigonometrical Ratios of 30°****Trigonometrical Ratios of 45°****Trigonometrical Ratios of 60°****Trigonometrical Ratios of 90°****Trigonometrical Ratios Table****Problems on Trigonometric Ratio of Standard Angle****Trigonometrical Ratios of Complementary Angles****Rules of Trigonometric Signs****Signs of Trigonometrical Ratios****All Sin Tan Cos Rule****Trigonometrical Ratios of (- θ)****Trigonometrical Ratios of (90° + θ)****Trigonometrical Ratios of (90° - θ)****Trigonometrical Ratios of (180° + θ)****Trigonometrical Ratios of (180° - θ)****Trigonometrical Ratios of (270° + θ)****Trigonometrical Ratios of (270° - θ)****Trigonometrical Ratios of (360° + θ)****Trigonometrical Ratios of (360° - θ)****Trigonometrical Ratios of any Angle****Trigonometrical Ratios of some Particular Angles****Trigonometric Ratios of an Angle****Trigonometric Functions of any Angles****Problems on Trigonometric Ratios of an Angle****Problems on Signs of Trigonometrical Ratios**

**11 and 12 Grade Math**

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