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Question

In the table, trigonometric ratios and the intervals of θ are given. Match the intervals with the ratios which are positive in those intervals.

θgives positive valuesp. (0,π2)1. Only sin θ,cosec θq. (π2,π)2. Only cos θ,sec θr. (π,3π2)3. Only tan θ,cot θs. (3π2,2π)4. All sin θ,cos θ,tan θ,cot θ,sec θ,cosec θ

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Solution

In first quadrant (0,π2)

We see, both x and y intercepts are positive. a>0,b>0

By definition of sinθ, cosθ, tanθ, cotθ, secθ, and cosec θ

All the trigonometric function gives positive values in first quadrant.

Second Quadrant (π2,π)

We see that x intercept in second quadrant is

negative. Therefore, a<0,b>0.

Functions which includes the values of

x - Intercept or adjacent will give negatives values.

All the other functions will give positive values.

By definition we know except sinθ & cosec θ , all the other functions use adjacent in their trigonometric ratios.

So, only sinθ & cosecθ give positive values in second quadrant.

Third quadrant (π,3π2)

We see both x & y - intercepts are negative

in III quadrant.

Therefore a<0,b<0

Both adjacent and opposite are negative.

So, trigonometric ratio which uses both adjacent and opposite gives positive value.

Only tanθ & cotθ use both adjacent and opposite.

In III quadrant, only tanθ & cotθ give positive values

Remaining trigonometric functions will give negative values.

Fourth quadrant (3π2, 2π)

xintercept is positive while y intercept negative.

Therefore a>0,b<0

Function which uses the values of y- intercept gives negative values, remaining functions give positive value.

Only cosθ & secθ doesn't will give positive value in fourth quadrant.


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