Value of tan 45°
Answer: The value of tan 45° is 1.
To find the value of tan45∘, we consider a right-angled triangle where one of the angles is 45°. In a 45-45-90 triangle, the two acute angles are equal, and the sides opposite these angles are also equal.
Let’s consider a triangle where the angle is 45°. If we denote the length of the side opposite the angle as a and the length of the adjacent side as b, in a 45-45-90 triangle, a = b.
The tangent of an angle (tan) is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.
tan 45° = opposite side/adjacent side = a/b = b/b = 1
Therefore, the value of tan45∘ is 1. This is a unique property of 45-45-90 triangles, where the ratio of the side opposite to the adjacent side is always 1.
Conclusion:
Understanding the value of tan 45° as 1 provides a fundamental insight into trigonometric ratios and their applications, particularly in right-angled triangles. In the context of a 45-45-90 triangle, where the two acute angles are equal, the tangent of 45° is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. By applying this definition and recognizing the unique properties of 45-45-90 triangles, we find that tan 45° equals 1.
Some Related Questions:
Can you explain the concept of a 45-45-90 triangle in detail?
45-45-90 triangle is a special type of right triangle where two of the angles are equal to 45 degrees each. In such triangles, the sides opposite the equal angles are congruent, resulting in unique relationships between the side lengths.
How does the value of tan 45° relate to the concept of slope in mathematics?
In mathematics, the tangent of an angle can also represent the slope of a line. When the angle is 45 degrees, the tangent value of 1 corresponds to a slope of 1, indicating that the line rises by 1 unit for every 1 unit of horizontal distance.