1. Geometry 9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles the relationship of sine and cosine in complementary angles

2. Complementary Angles add up to 90 degrees Complementary and Supplementary Angles Supplementary Angles add up to 180 degrees

3. In this diagram, we assume our angle is sitting on the positive x-axis and opening up toward the positive y-axis. Sine and Cosine As We Know Them…. i

4. Take a look at this pair of complementary angles and notice they have ray b in common. Sine and Cosine of a Complementary Angle a b c

5. Now imagine we have a pair of complementary angles making up a rectangle. Sine and Cosine of a Complementary Angle angle A angle B side d side c side b side a side e Notice the two triangles have side e in common.

6. Now imagine we have a pair of complementary angles making up a rectangle. Sine and Cosine of a Complementary Angle angle A angle B side d side c side b side a side e Notice that there are actually two angle As and angle Bs. angle B angle A

7. So every time we have a right triangle, we have a pair of complementary angles, even though they aren’t adjacent. Sine and Cosine of Complementary Angles angle A angle B From this diagram we know that angle A is 90°-B And we know angle B is 90°- A

8. So how do sine and cosine relate between complementary angles? Sine and Cosine of Complementary Angles Let’s label opposite, adjacent and hypotenuse to get a better picture angle X angle Y H O A H A O So the hypotenuse stays the same, but the opposite and adjacent sides switch

9. Using real numbers, let’s look at the difference between sine and cosine in complementary angles. Sine and Cosine of Complementary Angles angle X angle Y For angle X: Cos(X)= ⅘ Sin(X)= ⅗ Tan(X)=¾ For angle Y: Cos(Y)= ⅗ Sin(Y)= ⅘ Tan(Y)=4/3 5 4 3 Notice that Cos(X)=Sin(Y) Sin(X)=Cos(Y)

10. At your desk, find cosine, sine, and tangent of both complementary angles. Try This One! angle X angle Y 13 12 5

11. Answers At your desk, find cosine, sine, and tangent of both complementary angles. angle X angle Y 13 12 5 For angle X: Cos(X)=5/13 Sin(X)=12/13 Tan(X)=12/5 For angle Y: Cos(Y)=12/13 Sin(Y)=5/13 Tan(Y)=5/12