Ratios in Right Triangles WHAT YOU WILL LEARN To find trigonometric ratios using right triangles, and To solve problems using trigonometric ratios.

DEFINITIONS Trigonometry: The study of the properties of triangles and trigonometric functions and their applications. Trigonometric ratio: A ratio of the measures of two sides of a right triangle is called a trigonometric ratio.

Great Chief Soh Cah Toa Chief Soh Cah Toa. A young brave, frustrated by his inability to understand the geometric constructions of his tribe's battle dress, kicked out in anger against a stone and crushed his big toe. Fortunately, he learned from this experience, and began to use study and concentration to solve his problems rather than violence. This was especially effective in his study of math, and he went on to become the wisest man of his tribe. He studied many aspects of trigonometry; and even today we remember many of the functions by his name. When he became an adult, the tribal priest gave him a name that reflected his special nature -- one that reminded them of his great discoveries and of the event which changed his life. Because he was troubled throughout his life by the problematic foot, he was constantly at the edge of the river, soaking his aches in the cooling waters. For that behavior, he was named Chief Soh Cah Toa.

DEFINITIONS Sine: Opposite side over hypotenuse. Cosine: Adjacent side over hypotenuse. Tangent: Opposite side over hypotenuse.

Example Find the sin S, cos S, tan S, sin E, cos E and tan E. Express each ratio as a fraction and as a decimal. M Sin S = ME/SE = 3/5 or 0.6 Cos S = SM/SE = 4/5 or 0.8 4 3 Tan S = ME/SM = ¾ or 0.75 Sin E = SM/SE = 4/5 or 0.8 S E Cos E = ME/SE = 3/5 or 0.6 5 Tan E = SM/ME = 4/3 or 1.3

Example Find each value using a calculator. Round to the nearest ten thousandths. Cos 41 Sin 78

Example A plane is one mile about sea level when it begins to climb at a constant angle of 2 for the next 70 ground miles. How far above sea level is the plane after its climb? 2 h 1 mi Sea level 70 mi

Example Find m A in right triangle ABC for A(1,2), B(6,2), and C(5,4).

The End!