Question

Find tan0 where 0 is the angle shown. Give an exact value, not a decimal approximation. ​

Answer 1

`(3)/(√(55) )`

Explanation:

We require to find the third side of the triangle.

Using Pythagoras' identity.

The square on the hypotenuse is equal to the sum of the squares on the other 2 sides.

let x represent the third side, then

x² + 3² = 8², that is

x² + 9 = 64 ( subtract 9 from both sides )

x² = 55 ( take the square root of both sides )

x =
`√(55)`

Thus

tanΘ =
`(opposite)/(adjacent)` =
`(3)/(√(55) )`

Answer 2

tan(θ) = (3√55)/55

Explanation:

The third side (a) of the triangle is found from the Pythagorean theorem as ...

a^2 + b^2 = c^2

a^2 + 3^2 = 8^2

a^2 = 64 - 9 = 55

a = √55

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The mnemonic SOH CAH TOA reminds you that the tangent relation is ...

Tan = Opposite / Adjacent

Here, the side opposite angle θ has length 3, and the side adjacent is the one we just found, √55. Then ...

tan(θ) = 3/√55

Rationalizing the denominator by multiplying by (√55)/(√55), we find the tangent to be ...

tan(θ) = (3/55)√55