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Chapter 1 Right Triangle Ratios 1.1 Angles, Degrees, and Arcs 1.2 Similar Triangles 1.3 Trigonometric Ratios and Right Triangles 1.4 Right Triangle Applications 1.1 Angles, Degrees and Arcs Angles Degree measure of angles Angles and arcs Approximation of Earth’s circumference Approximation of the diameters of the sun and moon Angles Degree measure: An angle formed by one complete rotation of the terminal side in a counterclockwise direction has measure 360º. And angle with measure 1º is formed by 1/360 of the complete rotation. Right Triangle Ratios Special Angles Angle Measurement Degrees, minutes and seconds are used to measure angles. 1 degree = 60 minutes 1 degree = 3600 seconds 1 minute = 60 seconds Convert 12º6’23” to degrees: 6’ = (6/60)º and 23” = (23/3600)º 12º6’23” = (12 + 6/60 + 23/3600)º = 12.106º Angles and Arcs The circumference of a circle C, the length of an arc s, and the central angle subtended by the arc q are related by q / 360º = s / C Eratosthenes’ approximation: 7.5º/360º = 500 mi./C C = 24,000 miles Actual value today: 24, 875 miles 1.2 Similar Triangles Euclid’s Theorem and similar triangles Applications Euclid’s Theorem If two triangles are similar, their corresponding sides are proportional. Calculating Tree Height A tree casts a shadow of 32 ft. and a yardstick (3.0 ft.) casts a shadow of 2.2 ft. Find the height of the tree. Solution: x / 3 = 32 / 2.2 x = 3(32 / 2.2) x ≈ 44 ft. to the nearest integer 1.3 Trigonometric Ratios and Right Angles Pythagorean Theorem Trigonometric ratios Calculator evaluation Solving right triangles Pythagorean Theorem In a right triangle the side opposite the right angle is called the hypotenuse. If the other two legs are a and b and the hypotenuse is c, this relation is true: a2 + b 2 = c2 Trigonometric Ratios Complementary Relationships Solving Right Triangles Given the measures of two sides or of one side and an acute angle of a right triangle, the measures of the other sides and angles can be found. Example: 90º - 35.7º = 54.3º b = (124m) sin 35.7º ≈ 72.4m to three significant digits a = (124m) cos 35.7º ≈ 101m to three significant digits 1.4 Right Triangle Applications Mine shaft Length of air-to-air fueling hose Astronomy Mine Shaft Application tan q = opposite / adjacent tan 20º = x / 310 ft. x = (310 ft.) (tan 20º ) ≈ 110 ft. to three significant digits Air-to-Air Fueling Hose q = 32º and b = 120 ft. sec 32º = c/b c = 120ft. / cos 32º ≈ 140 ft. to three significant digits Astronomy sin 46º = x / 93,000,000 miles x = 93,000,000 sin 46º ≈ 67,000,000 miles to two significant digits