Question

Tan (90+Theta)

Answer 1

\[\tan (90 + \theta) = -\cot \theta\]

- sign is because tan(x) in second quadrant is negative...

Answer 2

@ waterineyes: You statement, "- sign is because tan(x) in second quadrant is negative..." is only true if theta is in first quadrant. You identity is true for theta in any quadrant because of the following reasons:
Review:
1) Supplementary Identity: tan (180 - theta) = - tan (theta)
2) Complementary Identity: tan (90 - theta) = cot (theta)
3) Negative Angle (Odd and even) Identity: tan (-theta) = -tan (theta)
Hence,
tan (90 + theta) = - tan [180 - (90 + theta)] = - tan (90 - theta) = - cot (theta)

Answer 3

@Mr._To,

You are saying if theta will be in first quadrant then my statement is true....

You are totally wrong...

In First Quadrant, all are Positive..

So, tan(90- theta) = cot(theta) (It is positive because value of tan in first quadrant in positive and not only tan all are positive in first quadrant)

Second thing if theta is in second quadrant, then value of tan in Second quadrant is negative, but sin and cosec in second quadrant is positive)..

So, tan(90 + theta) = -cot(theta) (cot is because there is 90 in the starting (for 90, 270 and all odd multiples of 90) we use complement, tan to cot, sin to cos, cos to sin, cot to tan etc etc))


And now tell me you said:

tan (90 + theta) = - tan [180 - (90 + theta)] = - tan (90 - theta) = - cot (theta)

Here, in second expression that is tan(180 - (90+theta)),,

In this, theta is in which quadrant???

Answer 4

@ waterineye
First, I apologize for the typos. It should have read your instead of you in "you statement" and "you identity".
Second,
You said, "- sign is because tan(x) in second quadrant is negative..." Please tell me in the identity
tan(90+θ)=−cotθ,
which angle is in the second quadrant, 90 + θ or θ?

Answer 5

Further, in the identity
tan(90+θ)=−cotθ,
which one has negative value, tan(90+θ) or −cotθ?

Answer 6

Firstly,
first answer of mine is: (90 + theta) is in second quadrant..

And tan is negative in second quadrant.. There is nothing to do with cot...

Answer 7

Since (90 + θ) is in Q2, θ is in Q1. That is what I meant.
My next question is:
If θ is in Q2, and thus (90 + θ) is in Q3, does the identity tan (90 + θ) = - cot (θ) hold?
Further, does the identity also hold if θ is in Q4 and (θ + 90) is in Q1?

Answer 8

Since in Q3, tan is positive, so tan(90+ theta) for theta in Q2, will be cot(theta)...

Answer 9

Now, if the theta is in Q4, then:

tan(90 + theta) = cot(theta) [Because (90 + theta) is in first quadrant if theta in Q4)..


If (theta + 90) is in Q1, then tan(90 + theta) = cot(theta)..
because here also, (90 + theta) is in Q1...

Answer 10

You said,
Now, if the theta is in Q4, then
tan(90 + theta) = cot(theta) [Because (90 + theta) is in first quadrant if theta in Q4)..

My question is
θ is in Q4, So cot (θ)<0. And (90 + θ) is in Q1, so tan (90 + θ)>0. Then why is
tan (90 + θ) = cot (θ) as you claimed?

Answer 11

What you want explain I am not getting that..
Say it in clean and clear words...

Answer 12

My point is that for all trigonometric identities, they hold regardless the value of θ or where the θ is on the unit circle. Your statement tan (90 + θ) = - cot (θ) is true without the remark tan (x) is negative since x is in Q2. Indeed ,the remark is redundant. Also, you didn't show your proof. Therefore, I added the proof and noted that the position of θ is irrelevant.
tan (90 +θ) = - tan [180 - (90 + θ)] (Supplementary Angles Identity)
= - tan (90 - θ) (Simplify)
= - cot (θ) (Complementary Identity)

Answer 13

See, by tanx, by x here I mean x to be (90 + theta) and not only theta..
x = (90 + theta)

So, if x is in 2nd Quadrant then tan(x) will be negative and = -cot(theta)..

Answer 14

Why not..
Sir, my math teacher has taught me with lot of efforts and I respect him a lot..
in the question:

tan(90+ theta), here if theta will be less than 90 then whole (90 + theta) will be in Q2..

and in Q2, tan is negative and as there is odd multiple that is 90, so we will take cot..
so, the final answer will be : - cot(theta)...

Answer 15

For simplicity, your teacher explained with assumption that θ is in Q2, so (90 + θ) is in Q2. This will take care the "sign" of tan (90 + θ) and cot (θ). However, you didn't prove why tan (90 + θ) = -cot (θ), so the proof is not complete.

We have the complementary angles Identity, which is
tan (90 - θ) = cot (θ)
However, we don't have the identity to show directly
tan (90 + θ) = - cot (θ)
Therefore, we have to use another identity, which is
tan (180 - θ) = -tan (θ)
Please remember that ALL trigonometric identities hold for all angles on the unit circle. You don't need to locate the angle(s) to prove the trigonometric identities.
Respecting your teacher is good.

Answer 16

No sir, theta is not in Q2, but the whole (90 + theta) is in Q2, that is what I am trying to say...

Answer 17

Typo. Yes, θ is in Q1. Please read my explanation, and ignore the quadrant because again, we don't need to consider where the angle is while proving trigonometric identities.