# 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.

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

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles 1. ### Measuring Capacity | Standard Unit of Capacity | Litre | Millilitres

Nov 29, 23 01:15 AM

We will discuss about measuring capacity. The milkman measures milk in liters. Petrol is given in liters. Mobil oil is sold in liters. Two milk bottles contain 1 liter of milk. One milk bottle

2. ### Addition and Subtraction of Units of Measurement | Metric Units

Nov 29, 23 12:54 AM

We can add the units of measurement like decimal numbers. 1. Add 5 m 9 dm and 11 m and 5 dm Solution: 5 m 9 dm = 5.9 m 11 m 5 dm = 11.5 m Hence, 5 m 9 dm + 11 m 5 dm = 17 m 4 dm or 17.4 m 2. Add 15 cm…