Certainly! To find the value of [tex]\(\tan 45^\circ\)[/tex], let's follow a step-by-step approach.
### Understanding 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.
### Special Angles:
45 degrees (or [tex]\(\frac{\pi}{4}\)[/tex] radians) is one of the special angles where the trigonometric functions have well-known values.
### Reference Triangle:
For a 45-degree angle, consider a right-angled isosceles triangle, where the two non-hypotenuse sides (legs) are of equal length.
1. Right-Angled Isosceles Triangle:
- If we take each leg of the triangle to be of length 1,
- Using the Pythagorean theorem, the hypotenuse [tex]\( h \)[/tex] would be:
[tex]\[
h = \sqrt{1^2 + 1^2} = \sqrt{2}
\][/tex]
2. Tangent of 45 Degrees:
- The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- In this case:
[tex]\[
\tan 45^\circ = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{1}{1} = 1
\][/tex]
### Conclusion:
Therefore, [tex]\(\tan 45^\circ = 1\)[/tex].
### Answer Options:
Given the options:
a. 0
b. 1
c. -1
d. [tex]\(\frac{1}{2}\)[/tex]
The correct answer is:
b. 1
