We have the expression 13 sin ⁡ [ tan −<!

Question

We have the expression

13 \sin [\tan^{- 1} (\frac{12}{5})]

.
Apparently the answer is 12, and I have to simplify it, and I'm assuming it means I have to show it's 12, without using a calculator.
Normally I show my own work in the questions, but in this case I have absolutely no clue how to. The only thing I know that might help is that

\tan (x) = \frac{\sin (x)}{\cos (x)}

Answer 1

There is a right triangle with side lengths 5, 12 and 13. Draw this triangle, and choose one of the two non-right angles t for which

\tan t = \frac{12}{5}

Recall that the tangent is the opposite side over the adjacent side.

Answer 2

Let

\tan^{- 1} \frac{12}{5} = θ

⟹ (i) \tan θ = \frac{12}{5}

and

(i i) - \frac{π}{2} \leq θ \leq \frac{π}{2}

⟹ \cos θ \geq 0

⟹ \cos θ = \frac{1}{\sec θ} = + \frac{1}{\sqrt{1 + \tan^{2} θ}} = \dots

⟹ \sin θ = \tan θ \cdot \cos θ = \dots

We have the expression 13 sin ⁡ [ tan −<!

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We have the expression 13 sin ⁡ [ tan −<!