20 ## TRIGONOMETRIC IDENTITIESTangent and cotangent identities AN IDENTITY IS AN EQUALITY that is true for any value of the variable. (An In algebra, for example, we have this identity: ( The significance of an identity is that, in calculation, we may replace either member with the other. We use an identity to give an expression a more convenient form. In calculus and all its applications, the trigonometric identities are of central importance. On this page we will present the main identities. The student will have no better way of practicing algebra than by proving them. Links to the proofs are below. Reciprocal identities
Again, in calculation we may replace either member of the identity with the other. And so if we see "sin θ", then we may, if we wish, replace it with ""; and, symmetrically, if we see "" then we may replace it with "sin θ". Problem 1. What does it mean to say that csc θ is the reciprocal of sin θ ? To see the answer, pass your mouse over the colored area. It means that their product is 1. sin θ csc θ = 1. Problem 2. Evaluate tan 30° csc 30° cot 30°.
Tangent and cotangent identities
The proof is here. Example 1. Show: tan θ cos θ = sin θ.
We begin:
We have arrived at the right-hand side.
These are called Pythagorean identities, because, as we will see in their proof, they are the trigonometric version of the Pythagorean theorem. The two identities labeled a
Problem 3. A 3-4-5 triangle is right-angled. a) Why? To see the answer, pass your mouse over the colored area. It satisfies the Pythagorean theorem. b) Evaluate the following:
Example 2. Show:
That is what we wanted to show.
Since these identities are proved directly from geometry, the student is not normally required to master the proof. However, all the identities that follow are based on these sum and difference formulas. The student should definitely know them. Here is the proof of the sum formulas. Example 3. Evaluate sin 15°. Example 4. Prove:
That is what we wanted to prove. Double-angle formulas There are three versions of cos 2α. The first is in terms of both cos α and sin α. The second is in terms only of cos α. The third is in terms only of sin α
That is what we wanted to prove.
—according to the previous identity with α = . Half-angle formulas The following half-angle formulas are inversions of the double-angle formulas, because α is half of 2α. The plus or minus sign will depend on the quadrant. Under the radical, the cosine has the + sign; the sine, the − sign.
on dividing both numerator and denominator by cos α. Products as sums
In the proofs, the student will see that the identities e) through h) are inversions of a) through d) respectively, which are proved first. The identity f) is used to prove one of the main theorems of calculus, namely the derivative of sin The student should not attempt to memorize these identities. Practicing their proofs -- and seeing that they come from the sum and difference formulas -- is enough. Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |