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Sine & Cosine of Complementary Angles

Sine & Cosine of Complementary Angles
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  • 0:02 Complementary Angles
  • 0:39 Right Triangle Trigonometry
  • 1:48 Sine & Cosine
  • 2:16 Examples
  • 3:33 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll look at the relationship between the sine and cosine of complementary angles. We'll discuss where this relationship comes from and explore examples of how it can be used.

Complementary Angles

Today we're going to talk about the relationship between sine and cosine when it comes to complementary angles. Complementary angles are simply two angles that add up to equal 90 degrees. For example, if angle A measures 37 degrees and angle B measures 53 degrees, they're complementary angles, because 37 + 53 = 90.

Right Triangle Trigonometry

When we take the sine and cosine of complementary angles, we find they have a special relationship. To understand this relationship, we're going to make use of some right triangle trigonometry. A right triangle has one angle measuring 90 degrees. Because of this and the fact that the angles of any triangle add up to 180 degrees, the other two angles of the triangle add up to 90 degrees. So, the other two angles are complementary.

sincoscomp2

In this image, we have a right triangle with the complementary angles A and B. To find the sine and cosine of an angle in a right triangle, we use the following rules:

  • Sine = opposite / hypotenuse
  • Cosine = adjacent / hypotenuse

To find the sine of an angle in a right triangle, we calculate the length of the side opposite of that angle divided by the length of the hypotenuse. Similarly, to find the cosine of an angle in a right triangle, we divide the length of the side adjacent to that angle by the length of the hypotenuse.

Now let's take a look at the sine and cosine of the angles in our right triangle:

sinA = a / H cosB = a / H
sinB = b / H cosA = b / H

Notice anything special in the table?

If you said that it looks like sinA = cosB and sinB = cosA, great job! That's exactly the observation that is going to explain the relationship between sine and cosine of complementary angles.

Sine and Cosine of Complementary Angles

We just saw that in our right triangle, sinA = cosB and sinB = cosA. Recall that A and B are complementary angles. This tells us that for two complementary angles, the sine of one equals the cosine of the other. It's pretty neat that we were able to derive this relationship from a bit of right triangle trigonometry.

Okay, now that we know the relationship and where it comes from, let's put it to practice!

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