Introduction to Trigonometry Lesson 9.9

What is Trigonometry? The shape of a right triangle is determined by the value of either of the other two angles. This means that once one of the other angles is known, the ratios of the various sides are ALWAYS the same regardless of the size of the triangle. These ratios are described by following “trigonometric functions” of the known angle. This means that if one angle and one side length is known, all other angles and side lengths can be determined. OR… it means that if two sides of the triangle are known, the third side and all other angles can be determined.

Three Trigonometric Ratios 1.Sine of  A = sin  A = opposite leg = hypotenuse 2.Cosine of  A = cos  A = adjacent leg = hypotenuse 3.Tangent of  A = tan  A = opposite leg = adjacent leg a b c C B A acac bcbc abab

Memorize this… S O H C A H T O A Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse Tangent Opposite Adjacent

S O H C A H T O A Memorize this… I N E PPOSITEPPOSITE YPOTENUSEYPOTENUSE OSINEOSINE DJACENTDJACENT YPOTENUSEYPOTENUSE ANGENTANGENT PPOSITEPPOSITE DJACENTDJACENT

Find cos  A 1.By the Pythagorean Theorem find side c. 2.c = 13 3.cos  A = adjacent leg to  A = hypotenuse Find tan  B 1.tan  B = leg opposite  B = leg adjacent to  B 12 5

Δ ABC is an isosceles triangle as marked. Find sin  C. 1.Draw in an altitude to make a right triangle. 2.Use the Pythagorean Theorem to find the length of the altitude. 3.AD = 20 4.Sin  C = opposite = hypotenuse 30 BC A =

Use the fact that tan 40º ≈ to find the height of the tree to the nearest foot. Tan 40 º = opposite = adjacent ≈ h (50) ≈ h ≈ h The tree is ≈ 42 feet tall. h 50

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