Introduction to Trigonometry NOTES 9.9 Introduction to Trigonometry

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 B a c A b C Sine of A = sin A = opposite leg = hypotenuse Cosine of A = cos A = adjacent leg = hypotenuse Tangent of A = tan A = opposite leg = adjacent leg a c b c a b

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

S O H C A H T O A OSINE ANGENT I N E Memorize this… DJACENT PPOSITE YPOTENUSE ANGENT I N E YPOTENUSE DJACENT

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

ΔABC is an isosceles triangle as marked. Find sin C. Draw in an altitude to make a right triangle. Use the Pythagorean Theorem to find the length of the altitude. AD = 20 Sin C = opposite = hypotenuse A 25 25 20 B C 15 15 30 20 = 4 25 5

h 50 Tan 40º = opposite = adjacent 0.8391 ≈ h 50 0.8391(50) ≈ h Use the fact that tan 40º ≈ 0.8391 to find the height of the tree to the nearest foot. h 50 Tan 40º = opposite = adjacent 0.8391 ≈ h 50 0.8391(50) ≈ h 41.955 ≈ h The tree is ≈ 42 feet tall.