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Common Core Standard   G.SRT.7 High School Geometry
right triangle complementary angles sin cosine relationship

Complementary angles sine cosine

Complementary angles of right triangles create a relationship between sin and cosine.

The sin is equal to the cosine of the complementary angle, and the cosine is equal to the sin of the complementary angle.

Sinθ=Cosine (90◦-θ) and Cosineθ = Sine (90◦-θ)
Sin of 60◦ = Cosine of 30◦ 
Sin of 60◦ = .86602 and Cosine 30◦ = .86602

The sin of 60 degrees = opposite over hypotenuse

The cosine of 30 degrees equals adjacent over hypotenuse.
Supplementary angles are  angles that add to 180 degrees.
32 + 58 = 90
23 + 67 = 90
40 + 140 = 180
90 + 90 = 180
22 + 68 = 90
31 + 59 = 90
55 + 35 = 90
supplementary angles 90 +90 =180
supplementary angles 40+140 =180
right triangle angles measures = 32 58 90
right triangle with angle measure 23 67 and right angle/definition complementary angle

In a right triangle because all three angle measures add to 180 degrees, the two non-right angles (acute angles) are complementary angles.

Definition of Complementary Angles (examples

Problem 5. What is the complementary angle of 47 degrees? Solution = 90 -47=43 degrees

Problem 6. What is the supplementary angle of 47 degrees? Solution = 180 - 47 = 137 degrees

Problem 7. What is the complementary angle of 45 degrees?
Solution 90-45 =45 degrees

Problem 8. What is the supplementary angle of 45 degrees?
Solution 180-45=135 degrees

Problem 1. An angle of  30° is the complement of ________ ?
30 + X  = 90
X  = 60°

Problem 3. An angle of 105° is the supplement of _____________ ?
105 + x = 180
x = 75°


This little symbol  ∠  is the symbol for angles just so you will recognize the symbol. The definition of complementary angle is two angles that add to 90. Thirty is the complement of what angle? Well the definition says “Take thirty minus the unknown angle and it will add up to ninety. So I will subtract thirty from both sides and I get sixty. So that is a sixty degree angle. OK now let’s look at of an application of complementary. In a right triangle the two acute angles are complementary. So if I know one angle is sixty eight I can subtract it from ninety and I get twenty two degrees. So that makes this angle twenty two so sixty eight and twenty two are complements of each other. Now let’s look at supplementary angle. The definition for supplementary angles is two angles that add to one eighty. So it is very similar, complementary is ninety, supplementary is one eighty. So a hundred and five degrees is the supplement of what angle. It’s the same thing. We will take X plus our one hundred and five and set it to one hundred and eighty subtract one hundred and five from both sides so X is seventy five degrees. So one hundred and five and seventy five are supplementary to each other. Now let’s look at an application problem of that concept. This is a straight line and straight lines have an angle of one hundred and eighty degrees. So when you draw an angle off of that line you will have two angles that are supplements of each other. So if this angle is sixty five what is this unknown angle? Well supplementary. I take one eighty and I subtract sixty five and I’m left with one fifteen, so my angle is one hundred and fifteen degrees. So let’s look at the rules for supplementary and complementary angles. Complementary are two angles that ad to ninety. Here is our example forty plus fifty is ninety. Forty and fifty are complements so they are complementary angles. Supplementary are two angles that add to one hundred and eighty so sixty and one twenty are supplements of each other. Hope this video was helpful.  

video tutorial definition complementary angles

Definition of Complementary Angles and Supplementary Angles


Sample Complementary Angle Problems

Definition Supplementary Angles


High School Geometry-Complementary Angles

Problem 2. What is the angle measure of X  ?

Step 1. 68 - x = 90

Step 2. x =  22°
Problem 4. What is the angle measure of X ?

Step 1. 65 - x = 180

Step 2. x =  115°
right triangle angle measure 68 degrees/definition complementary angle sample problem
angle measure
complementary angles of 55 and 35 degrees
complementary angles 3 and 59
complementary angles add to 90 22 + 68 =90

Video Tutorial Complementary Angles

Complementary angles are two angles that add to 90 degrees. If one angle is known then its complementary angle can be found by subtracting the angle from 90 degrees.