tan square theta by 2
Answers
Step-by-step explanation:
AN IDENTITY IS AN EQUALITY that is true for any value of the variable. (An equation is an equality that is true only for certain values of the variable.)
In algebra, for example, we have this identity:
(x + 5)(x − 5) = x2 − 25.
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
sin θ = 1
csc θ csc θ = 1
sin θ
cos θ = 1
sec θ sec θ = 1
cos θ
tan θ = 1
cot θ cot θ = 1
tan θ
Proof
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 " 1
csc θ "; and, symmetrically, if we see " 1
csc θ ",
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 ?
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It means that their product is 1.
sin θ csc θ = 1.
Lesson 5 of Algebra.
Problem 2. Evaluate tan 30° csc 30° cot 30°.
tan 30° csc 30° cot30 ° = tan 30° cot 30 ° csc 30 ° = 1· csc 30 ° = 2.
Topic 4.
Tangent and cotangent identities
tan θ = sin θ
cos θ cot θ = cos θ
sin θ
Proof
Pythagorean identities
a) sin2θ + cos2θ = 1.
b) 1 + tan2θ = sec2θ
c) 1 + cot2θ = csc 2θ
a') sin2θ = 1 − cos2θ.
cos2θ = 1 − sin2θ.
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') -- "a-prime" -- are simply different versions of a). The first shows how we can express sin θ in terms of cos θ; the second shows how we can express cos θ in terms of sin θ.
Note: sin2θ -- "sine squared theta" -- means (sin θ)2.
Problem 3. A 3-4-5 triangle is right-angled.
right triangle
a) Why?
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It satisfies the Pythagorean theorem.
b) Evaluate the following:
sin2θ = 16
25 cos2θ = 9
25 sin2θ + cos2θ = 1.
Example 1. Show:
trigonometric identities trigonometric identities
Solution: The problem means that we are to write the left-hand side, and then show, through substitutions and algebra, that we can transform it to look like the right hand side. We begin:
trigonometric identities trigonometric identities Reciprocal identities
trigonometric identities on adding the fractions
trigonometric identities Pythagorean identities
trigonometric identities
trigonometric identities Reciprocal identities
That is what we wanted to show.
Sum and difference formulas
sin (alpha + β) = sin alpha cos β + cos alpha sin β
sin (alpha − β) = sin alpha cos β − cos alpha sin β
cos (alpha + β) = cos alpha cos β − sin alpha sin β
cos (alpha − β) = cos alpha cos β + sin alpha sin β
Note: In the sine formulas, + or − on the left is also + or − on the right. But in the cosine formulas, + on the left becomes − on the right; and vice-versa.
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.