If tanθ = 1, then how to calculate θ = 45

[Indranil]

If If tanθ = 1, then how to calculate θ = 45? I am doing calculation below:

tanθ = 1, θ = 1/tan = 45 Am I correct?

 

[MarkFL]

Because of the periodicity of the tangent function, we may state:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$

If we choose $\displaystyle k=0$, then:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}=45^{\circ}$

 

[Deleted member 4993]

If If tanθ = 1, then how to calculate θ = 45? I am doing calculation below:

tanθ = 1, θ = 1/tan = 45 Am I correct?

Using what method?

You can prove this from various starting point and method.

One method would be to use unit circle.

Another would be to use the definition tan(Θ) = sin(Θ)/cos(Θ)

Mark showed you yet another method....

 

[HallsofIvy]

If If tanθ = 1, then how to calculate θ = 45? I am doing calculation below:

tanθ = 1, θ = 1/tan = 45 Am I correct?

No, you are not correct! In the first place "tanθ" is NOT "tan" times "θ", it is the tangent function applied to the variable θ. It would be better to write tan(θ). To solve tan(θ)= 1 you have to determine the value of the variable θ such that when the tangent function is applied to it, the result is 1. To find that θ you use the "inverse function", arctan().

tan(θ)= 1 so, applying arctan to both sides, arctan(tan(θ))= arctan(1). arctan(tan(θ))= θ (as long as θ is between $\displaystyle -]frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$) because that is how "arctan" is defined. So we have θ= arctan(1).

How do we determine what "arctan(1)" is? The trivial answer is that we use a calculator or computer! One of the things we have to do first is decide whether we want to work in "degrees" or "radians". In order to get 45 as an answer (and strictly speaking we should answer "45 degrees") we set the calculator in "degree mode". Using the "scientific calculator" that comes with Windows, I note that it comes up already in degree mode but that the trig functions given are "sin", "cos", and "tan" while I want the inverse function arctan. Clicking on the "up arrow" key, just below the \(\displaystyle \sqrt\) key, I now have "$\displaystyle sin^{-1}$", "$\displaystyle cos^{-1}$", and "$\displaystyle tan^{-1}$". (Those are simply an alternate notation for "arcsin", "arcos", and "arctan". I avoided using those because they can be confused with "sine, cosine, and tangent to the -1 power", the same kind of error you made when you tried to divide by "tan".).

Now enter "1" and click on the "$\displaystyle tan^{-1}$" key. That gives "45". But tangent is, as MarkFL said, periodic with period 180 degrees. So we can write the answer more generally as "45+ 180n degrees" where n can be any integer.

 

[Indranil]

Because of the periodicity of the tangent function, we may state:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$

If we choose $\displaystyle k=0$, then:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}=45^{\circ}$

'$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$' Could you please simplify the the step a little bit easier and also could you provide me with a picture so that I can understand well?

 

[JeffM]

'$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$' Could you please simplify the the step a little bit easier and also could you provide me with a picture so that I can understand well?

I am not going to draw you a picture because you will learn more by drawing it yourself and thinking about it.

Draw a right isosceles triangle.

It has two 45 degree angles. Why?

Let x be the length of that triangle's hypotenuse. What are the lengths of the other two sides? Why?

So what is the value of $\displaystyle sin ( 45^o )$? What is the value of $\displaystyle cos ( 45^o ).$?

$\displaystyle tan ( \theta ) = \dfrac{ sin( \theta)}{cos ( \theta )} \implies tan ( 45^o ) = WHAT?$

 

[Steven G]

If If tanθ = 1, then how to calculate θ = 45? I am doing calculation below:

tanθ = 1, θ = 1/tan = 45 Am I correct?

IF 1/tan = 45, tan = 1/45 Always. But this is all nonsense since, as Halls pointed out, tanθ is NOT tan*θ. tan has no meaning at all. However tanθ does have meaning, it is the ratio of the two sides of a right triangle whose angle is θ.

 

[Indranil]

I am not going to draw you a picture because you will learn more by drawing it yourself and thinking about it.

Draw a right isosceles triangle.

It has two 45 degree angles. Why?

Let x be the length of that triangle's hypotenuse. What are the lengths of the other two sides? Why?

So what is the value of $\displaystyle sin ( 45^o )$? What is the value of $\displaystyle cos ( 45^o ).$?

$\displaystyle tan ( \theta ) = \dfrac{ sin( \theta)}{cos ( \theta )} \implies tan ( 45^o ) = WHAT?$

1. Because we add three angles we get 180 degrees. But it is an isosceles triangle, it has two same sides and two same angles. if one is 90 degree then other ones would be 45 +45 = 90 degree.
2. I take the other sides as a and b because x² = a² +b ², x = √a² +b ² = a + b

3. sin(45°) = 1/√2 and cos(45°) = 1/√2

4. tan(45°) = sin(45°)/cos(45°) = (1/√2)/(1/√2) = 1

 

[Indranil]

Using what method?

You can prove this from various starting point and method.

One method would be to use unit circle.

Another would be to use the definition tan(Θ) = sin(Θ)/cos(Θ)

Mark showed you yet another method....

Could you discuss the unit circle method here, please?

 

[Dr.Peterson]

1. Because we add three angles we get 180 degrees. But it is an isosceles triangle, it has two same sides and two same angles. if one is 90 degree then other ones would be 45 +45 = 90 degree.
2. I take the other sides as a and b because x² = a² +b ², x = √a² +b ² = a + b

3. sin(45°) = 1/√2 and cos(45°) = 1/√2

4. tan(45°) = sin(45°)/cos(45°) = (1/√2)/(1/√2) = 1

Be careful there! It is not true that √(a² + b²) = a + b, as you are claiming; in particular, if a=1 and b=1, then x = √2 as you subsequently said, but a + b = 2. The radical does not distribute over the sum.

The rest of what you said is correct.

 

[Dr.Peterson]

'$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$' Could you please simplify the the step a little bit easier and also could you provide me with a picture so that I can understand well?

What part of this statement are you asking about? Is it finding that theta is pi/4 (the radian measure of 45 degrees)? Or the fact that the tangent function has period pi (180 degrees), so that you can add any integer multiple of that to find other solutions?

Could you discuss the unit circle method here, please?

Both of the issues above can be answered using different aspects of the unit circle.

Many people think of "the unit circle" simply as a visualization of the "special angles" and their trig functions; that is essentially just something to be memorized (you know that tan(45°) = 1, so that is a solution of tan(θ) = 1!).

But, more importantly, the unit circle is a way to define trig functions of any angle (not just acute angles), and so implies the period and allows you to find all solutions.

Your original question seems to be in the context of acute angles, so it may not be necessary for you to move beyond that yet. The important thing you need to learn, as others have mentioned, is that we don't solve this kind of equation by algebraic manipulations (dividing by "tan", for example, which is nonsense), but by either just knowing the answer, or using a calculator or table or, if necessary, very complicated calculations (such as are used by the calculator and the table compilers) to get the value.

If your textbook or course has not yet covered the unit circle, you can ignore it. But if you are interested, teaching the whole subject is really more than this site is meant to do for you. Find a textbook that explains it fully, or an online site that does the same, so you can get an orderly account. Just search the web for "unit circle trigonometry"!

 

[Indranil]

What part of this statement are you asking about? Is it finding that theta is pi/4 (the radian measure of 45 degrees)? Or the fact that the tangent function has period pi (180 degrees), so that you can add any integer multiple of that to find other solutions?



Both of the issues above can be answered using different aspects of the unit circle.

Many people think of "the unit circle" simply as a visualization of the "special angles" and their trig functions; that is essentially just something to be memorized (you know that tan(45°) = 1, so that is a solution of tan(θ) = 1!).

But, more importantly, the unit circle is a way to define trig functions of any angle (not just acute angles), and so implies the period and allows you to find all solutions.

Your original question seems to be in the context of acute angles, so it may not be necessary for you to move beyond that yet. The important thing you need to learn, as others have mentioned, is that we don't solve this kind of equation by algebraic manipulations (dividing by "tan", for example, which is nonsense), but by either just knowing the answer, or using a calculator or table or, if necessary, very complicated calculations (such as are used by the calculator and the table compilers) to get the value.

If your textbook or course has not yet covered the unit circle, you can ignore it. But if you are interested, teaching the whole subject is really more than this site is meant to do for you. Find a textbook that explains it fully, or an online site that does the same, so you can get an orderly account. Just search the web for "unit circle trigonometry"!

'$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$

If we choose $\displaystyle k=0$, then:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}=45^{\circ}$'
I don't understand this step. Please simplify the steps above so that I can understand well

 

[Dr.Peterson]

'$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}+\pi k=45^{\circ}+k\cdot180^{\circ}$ where $\displaystyle k\in\mathbb{Z}$

If we choose $\displaystyle k=0$, then:

$\displaystyle \tan(\theta)=1\implies \theta=\frac{\pi}{4}=45^{\circ}$'
I don't understand this step. Please simplify the steps above so that I can understand well

Which part of it? Replacing k with 0 is easy, and you have done it correctly; this gives the first and most obvious solution, in quadrant 1 (acute). I see nothing hard there.

Then, if you take k=1, you get 45 + 180 = 225 degrees, which is in quadrant 3. Can you see how this angle has the same tangent as 45 degrees?

 

[Indranil]

Be careful there! It is not true that √(a² + b²) = a + b, as you are claiming; in particular, if a=1 and b=1, then x = √2 as you subsequently said, but a + b = 2. The radical does not distribute over the sum.

The rest of what you said is correct.

'if a=1 and b=1, then x = √2 as you subsequently said, but a + b = 2. The radical does not distribute over the sum.' Could you simplify this part, please? I don't understand this part well.

 

[Dr.Peterson]

'if a=1 and b=1, then x = √2 as you subsequently said, but a + b = 2. The radical does not distribute over the sum.' Could you simplify this part, please? I don't understand this part well.

It will be easier if you can tell us what you think it means, so that I can correct it. As it is, I don't know what you are not understanding, because I can't tell what you are thinking. As a result, I don't know what you think is "simple" or not.

Do you really believe that √(a² + b²) = √a² + √b² = a + b? If so, then you are wrong; you are claiming that you can "distribute" a radical (root) over a sum, by saying that the root of a sum is equal to the sum of the roots.

I was correcting that statement, and then showing that for your specific example, it is not true: if a and b are 1, then √(a² + b²) = √2, but a + b = 2. They are not the same.

But, as I said, you didn't actually use this wrong claim, so you may not have meant it.

 

[Indranil]

Which part of it? Replacing k with 0 is easy, and you have done it correctly; this gives the first and most obvious solution, in quadrant 1 (acute). I see nothing hard there.

Then, if you take k=1, you get 45 + 180 = 225 degrees, which is in quadrant 3. Can you see how this angle has the same tangent as 45 degrees?

How do you get 'θ = π(pi)⁄4 + π(pi)κ'? Could you explain, please?

 

[Dr.Peterson]

How do you get 'θ = π(pi)⁄4 + π(pi)κ'? Could you explain, please?

Please don't write pi in both ways within an expression; that makes it unreadable. You mean θ = π/4 + πk, or you could write theta = pi/4 + pi k.

Given that tan(θ) = 1, we immediately know that one possible value for θ is π/4, because we have memorized the fact that tan(45°) = 1. If we forgot that, we would use the inverse tangent function on our calculator.

Then we know that adding any multiple of 180°, which is the period of the tangent function, leaves the tangent unchanged; so we can add kπ to our one known value of θ to obtain the other possible values.

Therefore, all possible solutions are π/4 + kπ, where k can be any integer. This means ..., -7/4 pi, -3/4 pi, 1/4 pi, 5/4 pi, 9/4 pi, ... are all solutions.

If you have more questions about this, please tell us what you are thinking, so we don't have to guess what we have to express differently. In particular, in your original question you didn't indicate whether you needed to find all solutions, or just the acute angle solution, so it is possible that all this is just beyond what you have learned, and you just have to keep working through your textbook until it is explained to you.