# Find the exact value of trigonometric function sec2x if sinx =3/5, cosx=4/5

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### 3 Answers

We have to find the exact value of sec 2x given that sin x = 3/5 and cos x = 4/5

Now sec x = 1/cos x. To find the value of sec 2x, we need to find cos 2x and then take its inverse.

The relation for cos 2x is (cos x)^2 - ( sin x )^2.

As cos x = 4/5 and sin x = 3/5, cos 2x = (cos x)^2 - ( sin x )^2 = (4 /5)^2 - (3/5)^2

=16/25 - 9/25

= 7/25

So we have cox 2x = 7/25

Therefore sec 2x = 1/ cos 2x = 25 / 7.

**The required value of sec 2x = 25/7.**

sinx = 3/5 and cosx = 4/5.

Since sinx = 3/5= 0.6 < sinpi/4 = 1/sqrt2 = sqrt2/2 = 0.70...

2x < pi/2

Since both sinx and cosx are positive , x is acute and in the first quadrant. 2x < pi/2 . Therefore cos2x is positive.

We know that sec 2x = 1/cos2x

sec2x = 1/ sqrt{2cos^2x-1}.

sec2x = 1/sqrt{2(4/5)^2 -1}

sec2x = 1/sqrt{2*16/25 -1}

sec2x = 5/sqrt{32-25}

sec2x = 5/sqrt 7.

Since sec 2x = 1/cos 2x, we'll have to determine the value of cos 2x, given the values of sin x and cos x.

cos 2x = cos (x+x)

cos 2x = cos x*cos x - sin x*sin x

cos 2x = (cos x)^2 - (sin x)^2

We'll substitute cos x and sin x by the given values:

sinx =3/5 and cosx=4/5

cos 2x = (4/5)^2 - (3/5)^2

cos 2x = 16/25 - 9/25

cos 2x = 7/25

We'll substitute the value of cos 2x in the formula of sec 2x:

sec 2x = 1/(7/25)

**The exact value of sec 2x = 25/7.**