# Inequalities (trig) + Attempted

• lovemake1
In summary, the conversation discusses finding solutions for the equation tanx + 3cotx = 4. The conversation also touches on finding solutions for tan = 1 and tan = 3, and representing tan^-1(3) in terms of radians. Some steps for finding solutions are provided, and it is clarified that tan^-1(3) cannot be written in terms of pi. Overall, the conversation focuses on solving the original equation and clarifying any doubts or confusion.
lovemake1

tanx + 3cotx = 4

## The Attempt at a Solution

Heres my attempt:

tanx + 3/tanx - 4 = 0
(tan^2x -4tanx + 3) / tanx = 0

tan^2x - 4 tanx + 3

(tanx - 3 ) (tanx - 1 ) = 0

tan = 3, tan = 1

im not sure about tan = 3, but for tan = 1 , pi/4 +180n,
is this correct? if it is how would i go about findin solutions for tan=3.

thanks.

The same way you solved tanx=1

tanx=3, the principle angle is tan-1(3)

how would we represent tan^-1(3) interms of radians ?
calculators are not allowed in our school, is there a way ?

lovemake1 said:
how would we represent tan^-1(3) interms of radians ?
calculators are not allowed in our school, is there a way ?

Unfortunately there is not.

hmm..
so we just find solutions for tan = 1 ?

lovemake1 said:
im not sure about tan = 3, but for tan = 1 , pi/4 +180n,
is this correct? if it is how would i go about findin solutions for tan=3.
"pi/4 +180n" -> Don't mix degree and radian measure. Also, state what "n" is. You should write
$$\frac{\pi}{4} + n \pi, n \in \mathbb{Z}$$

lovemake1 said:
hmm..
so we just find solutions for tan = 1 ?
No, you still need both solutions. The other could be written as
$$\tan^{-1} 3 + n \pi, n \in \mathbb{Z}$$

tan^-1(3) + npi
how do you write that interms of pi?

i've never seen anything like that. could you please give me an example?

lovemake1 said:
tan^-1(3) + npi
how do you write that interms of pi?

i've never seen anything like that. could you please give me an example?

you can't write tan-1(3) in terms of pi. That is why you have to leave it as is.

ok, just to make sure.. are my previous steps correct?
just to double check. thanks

lovemake1 said:
ok, just to make sure.. are my previous steps correct?
just to double check. thanks

Yes they are correct.

## 1. What are inequalities in trigonometry?

Inequalities in trigonometry are mathematical statements that compare two trigonometric expressions using symbols such as <, >, ≤, or ≥. These inequalities are used to represent the relationship between the values of different trigonometric functions, such as sine, cosine, and tangent, and can be solved using algebraic techniques.

## 2. How do you solve trigonometric inequalities?

To solve trigonometric inequalities, you can use algebraic techniques such as factoring, finding common denominators, and applying the properties of inequalities. You can also use the unit circle and trigonometric identities to simplify the expressions and solve for the unknown variable.

## 3. What is the difference between strict and non-strict inequalities in trigonometry?

In strict inequalities, the symbols used are < and >, and the values on either side of the inequality are not equal. In non-strict inequalities, the symbols used are ≤ and ≥, and the values on either side of the inequality can be equal.

## 4. How do you graph trigonometric inequalities?

To graph trigonometric inequalities, you can first graph the corresponding trigonometric functions. Then, you can shade the region that satisfies the given inequality. If the inequality is strict, the shading will be done with a dashed line, and if it is non-strict, the shading will be done with a solid line.

## 5. Can trigonometric inequalities be applied in real-life situations?

Yes, trigonometric inequalities can be applied in real-life situations, such as determining the maximum and minimum values of a function, finding the optimal angle for a structure, or calculating the range of possible values for certain physical quantities.

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