What is trigonometry? as we all know that what is meant by trigonometry is a branch of mathematics that studies the relationship of the sides and angles of a triangle and also the basic functions that arise because of the existence of these relations. Trigonometry is a comparison value defined in the cartesian coordinate of the elbow triangle.
Trigonometry has several functions, including sinus (sin), cosine (cos), tangent (tan), cosecan (cosec), secan (sec), and cotangen (cotan). Whereas the application of trigonometric identities in life is usually used to study science around astronomy , geography, and so on.
In general there are two or more trigonometric functions which, although they have different shapes, are graphically the same function. For example, two functionsand
which seems different, but both functions have graphs of trigonometric functions which can be described as follows.
So we can conclude that although the two functions look different, but actually the two trigonometric functions are the same. This means, for every x value,
This last equation is called the trigonometric identity formula, and we will discuss it in this discussion. The following figure lists eight basic trigonometric identities.
Note: The first three identities (in the orange color) graph of the trigonometric function are called inverse identities. The next two identities (in the green box) are called ratio identities. Meanwhile, the last three identities (in a blue box) are referred to as Pythagoras identity. The last two Pythagorean identities can be derived from the previous identity, namely cos² θ + sin² θ = 1, by dividing the two segments in a row with cos² θ and sin² θ. For example, by dividing both segments cos² θ + sin² θ = 1 with cos² θ, we get.
To derive the last Pythagorean identity, we must divide both cos² θ + sin² segments θ = 1 with sin² θ to get 1 + cot² θ = csc² θ.
After knowing the eight basic trigonometric identities above, we will then use those identities, together with our knowledge of algebra, to prove other identities.
Remember that trigonometric identity is a statement that contains the similarities of two forms for each variable change with the value in which the form is defined. To prove trigonometric identity, we use trigonometry substitution and algebraic manipulation with purpose.
Changing the shape on the left side of the identity into a shape like on the right side, or changing the shape on the right side of the identity to form as in the left side.
One thing that must be remembered in proving trigonometric identity is that we have to work on each segment separately. We must not use algebraic properties that involve both segments of identity — such as the sum of the two segments of the equation. Because, to do this, we must assume that the two segments are the same, which is something we will prove. In essence, we should not treat problems as an equation.
We prove trigonometric identity to build our ability to convert one form of trigonometric function into another. When we meet problems in other topics that require identity verification techniques, we usually find that the solution to the problem depends on how to change the form that contains trigonometry into a simpler form. In this case, we don’t have to always work with equations.
Ways to Prove Trigonometry Identity
- Usually it will be easier if we manipulate more complicated equation fields first.
- Look for a form that can be substituted with the trigonometric identity form in the trigonometric identity, so that it gets a simpler form.
- Pay attention to algebraic operations, such as fraction addition, distributive traits, or factoring, which might simplify the paths we manipulate, or at least guide us to simplified forms.
- If we don’t know what to do, change all trigonometric shapes to sine and cosine shapes. Maybe that can help.
- Always pay attention to the equation fields that we do not manipulate to ensure the steps we take towards the shape in the segment.
- In addition to the instructions above, the best way to become proficient at proving trigonometric identity is to practice a lot. The more trigonometric identities we have proven, then we will be more expert and confident in proving other trigonometric identities.
Examples of Trigonometric Identity Questions
Problem 1: Prove that sin θ cot θ = cos θ.
To prove this identity, we change the shape of the left segment to the shape of the right segment.
In this example, we change the shape on the left side to the shape on the right side. Remember, we prove identity by changing one form into another.
Problem 2: Prove that tan x + cos x = sin x (sec x + cot x).
Discussion We can begin by applying distributive properties to the right hand side to multiply the terms in parentheses with sin x. Then we can turn the right segment into an equivalent form and load tan x and cos x.
In this case, we change the right segment to the left side.
Before we go on to the next examples, let’s list some clues that might be useful in proving trigonometric identities. Some reviews of the Trigonometry Identity formula along with examples of trigonometric equations can be written this time. Hopefully what we have discussed in this article can be useful.