Trigonometry can be complicated for anyone to understand, even if you’re taking classes in the subject. As the name suggests, trigonometry is the study of triangles – more specifically, how the angles of a triangle relate to the lengths of its sides. Trigonometry is often used with circles and the unit 𝜋 (pi), as well.
The trigonometry problems you’ll see in this guide use the three primary trigonometric identities: sine (sin), cosine (cos) and tangent (tan). While sine, cosine and tangent are complex functions, most any scientific calculator will have sin, cos, and tan buttons to help you complete your trigonometry problems faster and easier.
The unit circle is another concept you’ll often see while studying trigonometry (and one that crops up in this guide as well). The unit circle is a circle with a radius of 1, and it illustrates the relationship between the angles and sides of a triangle as compared with the circle. We use the unit circle to make trigonometric calculations easier.
We can utilize the unit circle to figure out the value of sin and cos. This is because (cos 𝜽 = x) and (sin 𝜽 = y). For example, if you try and figure out sin(pi/3) unit circle, you’ll see that at the 60° mark, sin(pi/3) =√3/2 or 0.8660. You can find cos pi/3 and tan pi/3 with the unit circle as well.
In this quick guide, we’ll walk you through a few more equations involving pi, sin, cos, and tan on the unit circle.
What is sin(pi)?
If you look at the unit circle like we did in the last section, you’ll see that finding sine of pi is very easy – it’s just the number 0. However, in this section, we’ll go over finding the answer on a scientific calculator. Start by selecting your sine (or sin) button to start the operation, then use the 𝜋 button on your calculator to create sin(𝜋). Make sure to close the parentheses after 𝜋 if your calculator uses them. After pressing enter, you might be surprised to see that the answer is 0.0548, not 0.
The reason is that today’s calculators can’t represent pi fully, no matter how powerful they are. Because pi as we know it is a “floating-point number” – a number that cannot be represented accurately by a decimal – your calculator only uses the first few decimal values of pi in its calculations. Hence, using both the unit circle and a powerful calculator is so important. With the unit circle’s help, we know that sin(pi) is 0, even though our calculators can’t give us that exact answer.
The “angle in radians” is the angle that corresponds with your equation on the unit circle. Since sin(pi) is at the 180° mark on the unit circle, your equation for finding the angle in radians will be sin(180). By plugging sin(180) into your calculator, you will see that the answer is 0.
This is an easy one to answer, as we just looked at the degrees of sin(pi) in the paragraph above. Since 𝜋 is at the 180° mark, the answer to sin(pi) in degrees is 180°.
If you live in some parts of the world, you may not have heard of gradians (also known as grads or gon) before. Gradians are simply a metric representation of the degree system.
You can calculate grads by using this formula:
where r equals radians and g equals gradians. When values are inputed it will look like the below, meaning sin(pi) = 0 gradians.
Example 1. sin(pi)/3
We solved sin pi/3 up above, but don’t be fooled – sin(pi)/3 is a completely different equation. However, because of what we know about the unit circle (since we know sin(pi) = 0), sin(pi)/3 is also very easy to figure out as shown below. The value is 0.
However, because of how calculators work, our answer when using a calculator will be slightly different. Use the fraction function in your calculator to input as shown below. This simplifies to 0.0182.
What is inverse sin(pi)?
“Inverse” functions are another problem you will run into often while studying trigonometry. Inverse sin, inverse cos and inverse tan denote the inverse functions on most calculators as shown below. Arcsin, arccos, and arctan can also represent inverse functions.
To find inverse sin(pi), use the inverse sin button on your calculator:
Then type in 𝜋 and close the parentheses. Interestingly, your calculator will give you the answer “undefined.” This means that we cannot simplify inverse sin (pi) any further because the answer does not exist (or cannot be defined).
What is cos(pi)?
You can calculate cos(pi) in much the same way as you would sin(pi). Instead of using the sin button on your calculator, all you need to do is use the same process with the cos button.
To start, press cos on your calculator, then type in 𝜋 and close the parentheses. Keep in mind that because pi is a floating-point number, this function suffers from the same issues that sin(pi) does. Your calculator will give you the answer 0.9984. However, if we look back at the unit circle from before, we’ll see that this value is -1 instead.
To figure out the angle of cos(pi) in radians, we should look back at the unit circle again. 𝜋 is found at 180° on the unit circle, so this means our equation should be cos(180). If you plug this into your calculator, you’ll get the answer -1, just like the unit circle says.
Because we just looked at the unit circle, we already know that 𝜋 is located at the 180° mark. As such, cos(pi) in degrees equals 180°.
To figure out cos(pi) in gradians, we’ll use the same formula as before:
where r equals radians and g equals gradians. Since our answer to cos(pi) in radians was -1, our formula should look like the below:
If you plug this into your calculator, you’ll get the answer −63.6619.
What is inverse cos(pi)?
We’ll attempt to calculate inverse cos(pi) the same way we tried to figure out inverse sin(pi) before: by using the inverse cos function. Press the inverse cos button as shown below or arccos button on your calculator.
Then plug in 𝜋 and close the parentheses. Once again, we’ll receive the answer undefined, as inverse cos (pi) cannot be simplified any further on a calculator.
What is tan(pi)?
Tan(pi) works slightly differently than sin(pi) and cos(pi). Tan is the tangential line to the unit circle’s value, so while finding the answer with a calculator is the same as sin(pi) and cos(pi), finding it on the unit circle is slightly different.
To find tan(pi) on your calculator, just plug it in using the tan and 𝜋 buttons. You should get the answer 0.0548. To find the tangent on the unit circle, take the y-coordinate of the value and divide it by the x-coordinate.
Since this would equal 0/-1, we get the answer 0.
Since we already know that tan(pi) is at the 180° location on the unit circle, our formula should be tan(180). If you plug this into your calculator, you’ll once again get the answer 0.
Since we already know that 𝜋 is at the 180° mark, the answer to tan(pi) in degrees is 180°.
To convert tan(pi) to gradians, we will use the same formula as before:
where r equals radians and g equals gradians, so all we need to do is plug the radians (0) into the equation. Without even using our calculator, we should know that 0 multiplied by anything always equals 0, so our answer is 0.
What is inverse tan(pi)?
We can find the inverse of tan(pi) by using the inverse tan button on our calculator shown below.
By plugging 𝜋 into the equation then closing the parentheses, you’ll get the value 72.3432.
We hope this article has cleared trigonometry and the use of the unit circle up for you somewhat! Trigonometry and many other forms of math are not for the faint of heart, but with a little studying and practice, anyone can get the hang of it.
Do you have any questions that we weren’t able to answer in the sections above? If so, we may be able to do so here.
What is the unit of pi?
Believe it or not, there is no defined “unit” for pi because it is technically a ratio of the circumference to the diameter of any circle.
What is pi / 3 in degrees?
Pi/3 equals 60° in degrees.
What is pi / 3 angle in radians?
The angle of Pi/3 in radians is:
What is sin(-pi)?
If you plug sin(-pi) into your calculator, you’ll see that the answer is -0.0548.