The tangent is defined in right triangle trigonometry as the ratio of the opposite side to the adjacent side (It is applicable for acute angles only because it's only defined this way for right triangles). To find values of the tangent function at different angles when evaluating the tangent function, we first define the reference angle created by the terminal side and the x-axis. Then we calculate the tangent of this reference angle and determine whether it is positive or negative based on which quadrant the terminal side is in. In the first and third quadrants, the tangent is positive. In the second and fourth quadrants, the tangent is negative. The slope of the terminal side is also equal to the tangent.
Let us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of mathematics concerned with the application of specific functions of angles to calculations. In trigonometry, there are six functions of an angle that are widely used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their names and abbreviations. In relation to a right triangle, these six trigonometric functions. The sine of A, or sin A, is defined as the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) in a triangle. The other trigonometric functions are defined similarly. These functions are properties of the angle that are independent of the triangle's size, and measured values for several angles were tabulated before computers made trigonometry tables outdated. In geometric figures, trigonometric functions are used to calculate unknown angles and distances from known or measured angles. Trigonometry has a wide range of applications, from specific fields such as oceanography, where it is used to measure the height of tides in oceans, to the backyard of our home, where it can be used to roof a building, make the roof inclined in the case of single independent bungalows, and calculate the height of the roof etc. Here, we will discuss the tan theta formula in detail.
How to Find the Tangent?
You must first locate the hypotenuse to find the tangent. The hypotenuse is typically the right triangle's longest side. The next task is to decide the angle. There are only two angles to choose from. You cannot choose the right angle. After you've chosen an angle, you will mark the sides. The side opposite to this angle will be the opposite side and the side next to the angle is the adjacent side. After labelling the sides, you can take the required ratio. Let’s discuss ratios, what is tan theta and it’s practical applications?
What is Tan Theta?
The length of the opposite side to the length of the adjacent side of a right-angled triangle is known as the tangent function or tangent ratio of the angle between the hypotenuse and the base.
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As discussed, the tangent function is one of the three most common trigonometric functions, along with sine and cosine. The tangent of an angle in a right triangle is equal to the length of the opposite side (O) divided by the length of the adjacent side (A). It is written simply as 'tan' in a formula.
⇒ tan x = O/A
tan(x) is the symbol for the tangent function which is also called the tan x formula. It is one of the six trigonometric functions that are commonly used. Sine and cosine are most often associated with the tangent. In trigonometry, the tangent function is a periodic function that is very useful.
The tan formula is as follows:
⇒ Tan = Opposite/Adjacent
What is tan theta in terms of sine and cos?
⇒ tan x = sin x/cos x
or, tan theta = sin theta/cos theta (here, theta is an angle)
The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side, while the cosine of an angle is the ratio of the adjacent side to the hypotenuse side.
Hence, sin x = Opposite Side/Hypotenuse Side
cos x = Adjacent Side/Hypotenuse Side
Therefore, (tan formula) tan x = Opposite Side/Adjacent Side
Finding the Tangent of the Triangle
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Angles A and B are the two angles we will deal with in this triangle. To find our tangent, we must first find out hypotenuse. Our right angle is clear to me. Can you see it? That means our hypotenuse is direct across it, and the side that measures 5 is the hypotenuse. Can you see it? Okay, as we have our hypotenuse, let's choose an angle to work with. We'll choose angle B. As B is our angle, our opposite side is the side that measures 3. Our adjacent side is the one that measures 4 because it is the only side next to angle B which is not the hypotenuse.
This means that our tangent of angle B will be the ratio of the opposite side over the adjacent side or we can write it as 3/4 which will be equal to 0.75. Similarly, If we chose angle A, our sides will change and the tangent will be 4/3 which will be equal to 1.33.
Tangent to Find the Missing Side
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In a few problems, we will have to find the missing side of the triangle. This particularly has real-life applications like when construction companies are building on hills. In these kinds of problems, an angle is given to us. To solve these problems, we first have to locate the side that is missing. In the triangle given above, the side missed is the adjacent side to the angle given. Right? What we have to do now is just to write the equation from the definition of the tangent. Once we have the equation, we can move ahead, use algebra to solve for the variable which will be missing.
By multiplying both sides by x and then dividing both sides by tan 66, we've identified our variable. And by using the calculator to measure tan 66, we get the answer as 2.22.
Arctan - Inverse Tangent Function
There is an inverse function for any trigonometry function, for example, tan has arctan, which works in a reverse manner. These inverse functions have the same name as the originals, but with the word 'arc' added to the start. So, arctan is the opposite of tan. If we know the tangent of an angle and want to know the actual angle, we use the inverse function.
Large & Negative Angles
The two variable angles in a right triangle are always less than 90 degrees. However, we can find the tangent of any angle, irrespective of its height, as well as the tangent of negative angles. We can also graph the tangent function.
Tangents Will Be Used to Calculate the Height of a Building or a Mountain:
You can easily find the height of a building if you know the distance from where you observe it and also the angle of elevation. Similarly, you can find another side of the triangle if you know the value of one side and the angle of depression from the top of the house. All you need to know is a side and an angle of the triangle.
In fields like astronomy, mapmaking, surveying, and artillery range finding, trigonometry emerged from the need to compute angles and distances. Plane trigonometry deals with issues involving angles and distances in a single plane. Spherical trigonometry considers applications of related problems in more than one plane of three-dimensional space.