# Find the remaining side of the right angle triangle and the six trigonometric functions as functions of x : height (opp)= `sqrt(2x^(2)+8)`, hypotenuse = x-2. What is theta?

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The height of the right angle triangle is `sqrt(2x^2+ 8)` and the hypotenuse is x - 2. If the base is B,

`B^2 + (sqrt(x^2 + 8x))^2 = (x - 2)^2`

=> `B^2 + x^2 + 8x = x^2 - 4x + 4`

=> B^2 + 8x = 4 - 12x

=> B = `sqrt(4 - 12x)`

`theta = cos^-1(sqrt(4 - 12x)/(x - 2))`

`sin theta = sqrt(2x^2 + 8)/(x - 2)`

`tan theta = sqrt(2x^2 + 8)/sqrt(4 - 12x)`

The inverse of sine, cosine and tangent of the angle gives the cosec, sec and cotangent of the angle.