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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

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

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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

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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

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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.

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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

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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

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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

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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)

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

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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

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