[G.SRT.7] Similarity, Right Triangles, and Trigonometry #7

Objective

Common Core Text:

• • [G.SRT.7] Explain and use the relationship between the sine and cosine of complementary angles.

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Explanation

Here we have an ordinary right triangle. We're wondering if there are any relationships between sine (sin) and cosine (cos). Let's do some trig.

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For angle B

Sine of an Angle =

sinB =

sinB = = 0.6

Cosine of an Angle =

cosB =

cosB = = 0.8

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For angle C

sinC =

sinC = = 0.8

cosC =

cosC = = 0.6

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Interesting. You can see that the sine of B (sinB) is equal to the cosine of C (cosC). Also the sine of C (sinC) is equal to the cosine of B (cosB).

• sinB = cosC

• sinC = cosB

Will it always be this way? Yes. Why? Because

• whatever is opposite for one angle will be adjacent for the other angle. And because of that,

• whatever is sine for one angle will be cosine for the other angle.

That's the important part, but we can take it one step further. In this picture we used a picture of a triangle. In triangles, the angles of the two legs will always sum to 90° (complimentary angles), because the other angle is a right angle and all three need to add up to 180°. However, sines and cosines can be used with any angle, triangle or no triangle. But this relationship between sine and cosine will still work, as long as the two angles are complimentary, just like they would be in a triangle. Let's show this with a picture

Just like last time, looking at the triangle

• sinA = cosB

• sinB = cosA

We also have to angles that aren't part of a triangle. But since they are complimentary (50 + 40 = 90), the relationship will still hold

• sinE = cosG

• sinG = cosE

This is kind of a weird thing to remember. Here's my advice. Just remember that trigonometric ratios (sin, cos, tan) were invented for triangles. If you happen to have two complimentary angles A and B, sinA = cosB because A and B are acting like angles from a triangle.

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Practice your skills