Answer :
To determine the value of [tex]\(\tan 45^{\circ}\)[/tex], we need to evaluate the trigonometric function tangent for the angle of 45 degrees.
1. Recognize the angle and its position:
- The angle [tex]\(45^{\circ}\)[/tex] is in the first quadrant, where all trigonometric functions are positive.
2. Understand the properties of tangent:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.
- For a [tex]\(45^{\circ}\)[/tex] angle in a right triangle, both the opposite and adjacent sides are equal. Hence, their ratio is 1.
3. Evaluate [tex]\(\tan 45^{\circ}\)[/tex]:
- Since [tex]\(\tan 45^{\circ}\)[/tex] is the ratio of two equal sides, this gives [tex]\(\tan 45^{\circ} = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1\)[/tex].
4. Answer:
- Therefore, [tex]\(\tan 45^{\circ} = 1\)[/tex].
Given the options:
A. [tex]\(\frac{1}{2}\)[/tex]
B. [tex]\(\sqrt{2}\)[/tex]
C. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
D. 1
The correct answer is:
D. 1.
1. Recognize the angle and its position:
- The angle [tex]\(45^{\circ}\)[/tex] is in the first quadrant, where all trigonometric functions are positive.
2. Understand the properties of tangent:
- The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.
- For a [tex]\(45^{\circ}\)[/tex] angle in a right triangle, both the opposite and adjacent sides are equal. Hence, their ratio is 1.
3. Evaluate [tex]\(\tan 45^{\circ}\)[/tex]:
- Since [tex]\(\tan 45^{\circ}\)[/tex] is the ratio of two equal sides, this gives [tex]\(\tan 45^{\circ} = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1\)[/tex].
4. Answer:
- Therefore, [tex]\(\tan 45^{\circ} = 1\)[/tex].
Given the options:
A. [tex]\(\frac{1}{2}\)[/tex]
B. [tex]\(\sqrt{2}\)[/tex]
C. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
D. 1
The correct answer is:
D. 1.
