What is the exact value of the tangent of 45?

The tangent of 45 degrees is a commonly asked question when it comes to trigonometry. To ascertain the exact value of this trigonometric function, we need to utilize our knowledge of angles and their ratios.

First off, let's recall the definition of the tangent function. In trigonometry, the tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the adjacent side. Mathematically, tan(x) = sin(x)/cos(x). This formula helps us determine the value of the tangent for any given angle, including 45 degrees.

Now, let's refer to the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It plays a vital role in trigonometry as it allows us to relate angles and trigonometric functions.

When we look at the unit circle, we see that the angle 45 degrees is located in the first quadrant. In this quadrant, both the x-coordinate and y-coordinate of a point on the circle are positive. For a 45-degree angle, the point lies on the line y = x, which intersects the unit circle at the coordinates (1/√2, 1/√2).

Now, let's calculate the sine and cosine values for a 45-degree angle using the coordinates we obtained. For a 45-degree angle, the x-coordinate is cos(45) = 1/√2 and the y-coordinate is sin(45) = 1/√2.

Substituting these values into the tangent formula, we get tan(45) = sin(45)/cos(45) = (1/√2)/(1/√2) = 1/1 = 1. Therefore, the exact value of the tangent of 45 degrees is 1.

To summarize, one of the key trigonometric ratios, the tangent of 45 degrees, has an exact value of 1. Understanding the trigonometric functions and their definitions allows us to find accurate solutions to various mathematical problems.

What's the tangent of 45?

The tangent function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the lengths of its sides. When we talk about the tangent of an angle, we are referring to this ratio.

In this case, we are interested in finding the tangent of the angle 45 degrees. To do that, we need to know the lengths of the two sides of the right triangle that form this angle.

In a right triangle, the side opposite to the angle is called the opposite side, and the side adjacent to the angle is called the adjacent side. In our case, these two sides have the same length, so we can consider them as equal.

To find the tangent of 45 degrees, we need to divide the length of the opposite side by the length of the adjacent side. Since these two sides are equal, this division simplifies to dividing the length of one side by itself.

The tangent of 45 degrees, then, is equal to 1. This means that the opposite side and the adjacent side have the same length in a right triangle with a 45-degree angle.

The tangent function is widely used in various fields such as physics, engineering, and mathematics. It helps in solving problems involving angles and distances, as well as in modeling various natural phenomena.

Knowing the tangent of an angle can be helpful in calculations involving slopes, such as determining the angle of a ramp or the incline of a hill.

In conclusion, the tangent of 45 degrees is 1. This simple ratio tells us that the opposite side and adjacent side of a right triangle with a 45-degree angle have equal lengths. Understanding the tangent function is essential for solving trigonometric problems and applications in a variety of fields.

What is the value of tan if the angle is 45?

Tan, short for tangent, is a trigonometric function that relates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. The value of tan can be calculated by dividing the length of the opposite side by the length of the adjacent side.

When the angle is 45 degrees, the length of the opposite side and the length of the adjacent side are equal. This is because a 45-degree angle forms an isosceles right triangle, where the two legs are congruent. Therefore, if we divide the length of the opposite side by the length of the adjacent side, we get a ratio of 1.

So, in the case of a 45-degree angle, the value of tan is 1. This means that the lengths of the opposite and adjacent sides are equal, resulting in a tangent ratio of 1.

Knowing the value of tan for different angles is important in solving various mathematical and engineering problems that involve angles and triangles. The value of tan can be used to find missing side lengths, determine angles in a triangle, and solve trigonometric equations.

In summary, when the angle is 45 degrees, the value of tan is 1. This indicates that the lengths of the opposite and adjacent sides of the right triangle are equal.

Why is tan 45 equal to 1?

The tangent function, commonly abbreviated as tan, is a mathematical function that relates the angles of a right triangle to the ratios of its sides. When we consider the angle of 45 degrees, which is a well-known angle in geometry, we can explore why the value of tan 45 is equal to 1.

The angle of 45 degrees is special because it divides the right triangle into two congruent 45-45-90 triangles. In these triangles, the two legs are of equal length, and the hypotenuse is always the square root of 2 times the length of the legs.

Now, let's consider one of these 45-45-90 triangles. The measure of each acute angle, including 45 degrees, is equal to 45/2 = 22.5 degrees. By using trigonometric functions, we can determine the ratio of the length of the opposite side to the length of the adjacent side, which is the definition of the tangent function (tan).

For a 45-degree angle, the opposite and adjacent sides are of equal length, as the two legs of the triangle are congruent. Therefore, the ratio of the length of the opposite side to the length of the adjacent side is 1/1, which simplifies to 1.

Therefore, tan 45 is equal to 1. This means that the tangent of a 45-degree angle is the same as the ratio of the length of the opposite side to the length of the adjacent side, which is 1. It is a fundamental geometric property that holds true for all right triangles with a 45-degree angle.

What is tan 45 without calculator?

What is tan 45 without calculator?

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One important trigonometric ratio is the tangent (tan) function, which relates the opposite side of a right triangle to its adjacent side.

When we talk about tan 45, we are referring to the tangent of a 45-degree angle. To find the tangent of 45 degrees without using a calculator, we need to remember a few key pieces of information.

First, it is important to know that tan 45 is equal to the ratio between the opposite side and the adjacent side of a 45-degree right triangle. In a 45-degree triangle, the opposite side and the adjacent side have the same length.

Therefore, if we let the length of the opposite side be x and the length of the adjacent side be x, the ratio of tan 45 can be expressed as x / x, which simplifies to 1.

This means that the tangent of 45 degrees is equal to 1. Without a calculator, we can confidently state that the value of tan 45 is 1.

In conclusion, the tangent of 45 degrees without a calculator is 1. This value is derived from the fact that in a 45-degree right triangle, the opposite side and the adjacent side have the same length.