Question

find the exact value of tan 45°.
tan 45° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of tangent

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side, i.e., $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. For a $45^\circ - 45^\circ - 90^\circ$ triangle, the two legs (opposite and adjacent to the $45^\circ$ angle) are equal in length. Let the length of each leg be $a$.

Step2: Calculate $\tan 45^\circ$

Using the definition of tangent, for $\theta = 45^\circ$, we have $\tan 45^\circ=\frac{a}{a}$ (since opposite and adjacent sides are both length $a$). Simplifying $\frac{a}{a}$ (where $a
eq0$) gives $1$. Also, from the unit circle, $\tan\theta=\frac{\sin\theta}{\cos\theta}$. We know that $\sin 45^\circ=\frac{\sqrt{2}}{2}$ and $\cos 45^\circ=\frac{\sqrt{2}}{2}$. So $\tan 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

Answer:

$1$