Complementary Angles Calculator
With this complementary angles calculator, you can easily find out what the complementary angle is for your given one. Furthermore, you can quickly check if two angles are complementary to each other – just by inputting two angles in degrees or radians. If you're not sure what complementary angles are, make sure to first read the definition and explanation on how to find complementary angles.
Are you ready to dig into the world of geometry? This complementary angles calculator is only one of our many awesome tools – maybe you want to look at the problem from a different angle? :) If so, then our angle conversion tool can help with that before you continue reading.
What are complementary angles? Complementary angles definition
The complementary angles definition is really straightforward – complementary angles are two angles which add up to 90°. And that's pretty much it!
Additionally, if the angles are adjacent (but they don't need to be!), they will form a right angle:
You can check out yourself whether or not two angles are complementary – you can do it manually or enter the values into our complementary angles calculator.
How to find complementary angles?
Now that you know what complementary angles are, it's a no-brainer. First, you need to know what you want to find:
I want to find a complementary angle to my given angle
Subtract your angle from 90°:
complementary angle = 90° - angle; or in π rad
complementary angle = π/2 - angle
I want to find out if two angles are complementary
Check if the sum of two angles equals 90° (π/2):
angle1 + angle2 = 90° (π/2)– the angles are complementary; or
angle1 + angle2 ≠ 90° (π/2)– the angles are not complementary.
Of course, you can simply use our complementary angle calculator. It's here for you.
Non-adjacent and adjacent complementary angles
When learning geometry, you will often find adjacent complementary angles, even though you don't notice or pay much attention to them!
- Look at the diagonal of the rectangle: it splits the right angle into two adjacent complementary angles. The same works for a special type of rectangle – a square – here, the angle and its complementary angle are even equal to each other (45°).
- Take a look at a right triangle's height – did you notice that altitude, h꜀, from the right angle to the hypotenuse creates two similar triangles? The adjacent complementary angles also have the same measures as the angles in the original right triangle.
There are analogical situations for any other shape that has a right angle.
And what is an example of some non-adjacent complementary angles?
Well, just think about a right triangle again – the two acute angles of such a triangle are complementary. Why? Because the sum of the angles in a triangle equals 180°, and by subtracting the right angle, we're left with 90° for the two remaining angles.
Complementary and supplementary angles
A similar concept to complementary angles is a supplementary angle (yes, we have a supplementary angles calculator too). This time, two angles must sum up to 180° to be supplementary, not to 90° like in complementary angles.
How to remember which is which? There are plenty of mnemonic techniques, such as:
Supplementary angles sum to 180°, so to the ⎯ straight line
Complementary angles add up to a right angle, 90°: it is a ⊾ corner, right? Also, you can remember it with the phrase "it's right to say compliments" (be mindful about the spelling though, it's complementary angle, not complimentary)
Another way of remembering it is that the letter c comes before s in the alphabet, the same way that a complementary angle (90°) is smaller than a supplementary angle (180°).
If you're a visual learner, try to sketch it: for example, you can draw the C as a 9 and the S as an 8:
Complementary angles relations
Dealing with the trigonometric functions of two angles may be pretty tough, unless you know the relationship between them. So what happens if you know that α and β are supplementary angles? Then, as
α + β = 90° (π/2) we can state that:
The sine of an angle equals the cosine of its complement, and vice versa:
sin(α) = cos(90° - α) = cos(β)
cos(α) = sin(90° - α) = sin(β)
The tangent of an angle is the reciprocal of the tangent of its complement:
tan(α) = tan(90° - β) = 1/tan(β)
in other words, you can say that the multiplication of the tangents of complementary angles equals one:
tan(α) × tan(β) = 1
The general rule to remember is that the trigonometric function of an angle is equal to the cofunction of its complement!
function & cofunction
in π rad
sin(α) = cos(90°-α) = cos(β)sin(α)
=cos(π/2 - α) = cos(β)
cos(α) = sin(90° - α) = sin(β)sin(α)
=sin(π/2 - α) = sin(β)
tan(α) = cot(90° - α) = cot(β)tan(α)
=cot(π/2 - α) = cot(β)
cot(α) = tan(90° - α) = tan(β)cot(α)
=tan(π/2 - α) = tan(β)
sec(α) = csc(90° - α) = csc(β)sec(α)
=csc(π/2 - α) = csc(β)
csc(α) = sec(90° - α) = sec(β)csc(α)
=sec(π/2 - α) = sec(β)