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To achieve mastery of this lesson, make sure that you develop responses to the essential questions listed below:
How do similar right triangles lead to the definitions of the trigonometric ratios?
What is the relationship between the sine and cosine of complementary angles?
GeOverview
Take a minute to refresh your memory a bit on what you just learned.
The term trigonometry comes from a Greek word meaning “triangle measuring.”
The sine of an angle within a right triangle is found by dividing the length of the opposite side by the length of the hypotenuse.
The cosine of an angle within a right triangle is found by dividing the length of the adjacent side by the length of the hypotenuse.
The tangent of an angle within a right triangle is found by dividing the length of the opposite side by the length of the adjacent side.
Right triangle DEF is shown. Angle E is the right angle. Segment DE is 3. Segment EF is 4. Segment FD is 5. angle D is highlighted.
sin ∠D = opposite over hypotenuse = four over five
cos ∠D = adjacent over hypotenuse = three over five
tan ∠D = opposite over adjacent = four over three
Complementary angles have special relationships with the sine and cosine trigonometric functions:
The sine of an angle has the same value as the cosine of the complementary angle.
The cosine of an angle has the same value as the sine of the complementary angle.
Two nested right triangles are shown. The smaller nested right triangle is a 5, 12, 13 right triangle and the larger right triangle is labeled 10, 24, 26.
In the image to the right, two nested right triangles are shown with angle Θ marked by point C. Similar right triangles and their side ratios lead to the properties and definitions of the trigonometric ratios.
Small triangle:
vertical over hypotenuse = 12 over 13
Large triangle:
vertical over hypotenuse = 24 over 26 = 12 over 13
Two nested right triangles are shown. The smaller nested right triangle is a 5, 12, 13 right triangle and the larger right triangle is labeled 10, 24, 26.
Notice the relationship between the first ratios and sin Θ ratio.
Small triangle:
sin Θ = opposite over hypotenuse = 12 over 13
Large triangle:
sin Θ = opposite over hypotenuse = 24 over 26 = 12 over 13
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