Geometry B, Unit 10 (All lessons) 100% Pass

Geometry B, Unit 10 (All lessons) 100% Pass geometric mean the positive nth root of the product of n factors inclinometer a device for measuring the amount of incline or tilt of an object or a surface trigonometry the branch of mathematics that deals with the relationships between the sides and the angles of triangles Theorem 10.1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. Corollary 1 of Theorem 10.1 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. Corollary 2 of Theorem 10.1 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Prove: ΔABC~ΔCDB~ΔADB 3. Def. of Altitude 5. Reflexive 6. AA 7. Reflexive 8. AA 9.Transitive A(n) ___ is a device for measuring the amount of incline or tilt of an object or a surface. inclinometer The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Cor. 2 of Thrm. 10.1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. Thrm. 10.1 The branch of mathematics that deals with the relationships between the sides and the angles of triangles is called ___. trigonometry The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. Cor. 1 of Thrm. 10.1 The ___ is the positive nth root of the product of n factors. geometric mean Find the arithmetic mean and geometric mean of the set of numbers.{2, 18} Arithmetic mean: 10 Geometric mean: 6 Find the arithmetic mean and geometric mean of the set of numbers. {2, 32} Arithmetic mean: 17 Geometric mean: 8 Greek mathematician and teacher who lived in the 2nd century B.C. Hipparchus useful for finding the average of a set of values that are similar arithmetic mean used to compare values that are proportional trig mean If x=2 cm; y=8 cm then find the length of the altitude h. 4 cm If h=6 in; y=18 in then find x. 2 in If c=10 cm; x=1.6 cm; find s1 . 4 cm If c=10 cm; y=8.4 cm; find s2 . 9.17 cm cosecant a trigonometric ratio consisting of the length of the hypotenuse to the length of the side opposite a given angle of a right triangle; the reciprocal of the sine cosine a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of the hypotenuse cotangent a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of side opposite the given angle; the reciprocal of the tangent reciprocal one of a pair of values whose product is one; also called the multiplicative inverse secant a trigonometric ratio consisting of the length of the hypotenuse to the length of the side adjacent to a given angle of a right triangle; the reciprocal of the cosine sine a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the hypotenuse tangent a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the side adjacent to the given angle trigonometric ratios the ratios of the lengths of the two sides of a right triangle Mnemonic A mnemonic device that is commonly used to help students remember the primary trigonometric functions is SOHCAHTOA, pronounced [soh-kuh-TOH-uh]. It stands for: SOH = Sine - Opposite over Hypotenuse CAH = Cosine - Adjacent over Hypotenuse TOA = Tangent - Opposite over Adjacent Or, you could use this one: Some Old Horses Can't Always Hide Their Old Age. ___ ratios are the ratios of the lengths of the two sides of a right triangle. Trigonometric One of a pair of values whose product is one is called a(n) ___; also called the multiplicative inverse. reciprocal The ___ is a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the hypotenuse. sine ___ is a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of the hypotenuse. Cosine ___ is a trigonometric ratio consisting of the length of the side opposite a given acute angle of a right triangle to the length of the side adjacent to the given angle. Tangent ___ is a trigonometric ratio consisting of the length of the side adjacent to a given acute angle of a right triangle to the length of side opposite the given angle; the reciprocal of the tangent. Cotangent ___ is a trigonometric ratio consisting of the length of the hypotenuse to the length of the side adjacent to a given angle of a right triangle; the reciprocal of the cosine. Secant ___ is a trigonometric ratio consisting of the length of the hypotenuse to the length of the side opposite a given angle of a right triangle; the reciprocal of the sine. Cosecant Match the trigonometric name with the correct ratio. sin opposite / hypotenuse Match the trigonometric name with the correct ratio. cos adjacent / hypotenuse Match the trigonometric name with the correct ratio. tan opposite / adjacent Find sin D. 3/5 Find tan D. 3/4 Find cos F. 3/5 Find tan F. 4/3 Find cos D. 4/5 Find sin F. 4/5 solving a triangle calculating unknown side lengths or angle measures of a triangle if certain of the side lengths and/or angle measures are known Which statement is NOT correct? If you know the measure of all three angles of a right triangle, you can find the length of each of the three sides. The process known as _____ a triangle is used for calculating unknown side lengths or angle measures of a triangle if certain of the side lengths and/or angle measures are known. solving The three expressions, sin-1, cos-1, and tan-1 are called _____ trig functions and are used to find the measure of the acute angles of a right triangle if you know the lengths of at least two sides. inverse Find the length of c . Use the values sin30∘=0.5,cos30∘=0.866,tan30∘=0.577 1a. e 1b. i 2. a 3. l 4. g 5. j Find the length of b . Use the values sin30∘=0.5,cos30∘=0.866,tan30∘=0.577 . 1a. h 1b. a 2. j 3. l 4.g 5. f Find the measure of ∠A . 1a. e 1b. i 2. j 3. d 4. f angle of depression the angle formed by a horizontal line and a line of sight to a point below the horizontal angle of elevation the angle formed by a horizontal line and a line of sight to a point above the horizontal Angle of ___ is the angle formed by a horizontal line and a line of sight to a point below the horizon. depression Angle of ___ is the angle formed by a horizontal line and line of sight to a point above the horizon. elevation Given the angle of elevation and distance from the Eiffel tower, find its height (x) . 1a. k 1b. j 2. d 3. i 4. b 5. f The hot air balloon is 500 feet off the ground. The observer sees his landing zone at an angle of depression of 45°. Find the horizontal distance to his landing spot. 1a. d 2. j 3. b 4. e 5. f Law of Sines for any ABC with side lengths a, b, and c: sin A/a = sin B/b = sin C/c Law of Cosines for any ABC with side lengths a, b, and c: a^2 = b^2 + c^2 - 2bcCOS A b^2 = a^2 + c^2 - 2acCOS B c^2 = a^2 + b^2 - 2abCOS C 1) Choose the law and the formula that would be used to solve the triangle. Law: Law of Sines Formula: sinB / b = sinC / c 2) Choose the law and the formula that would be used to solve the triangle. Law: Law of Sines Formula: sinA / a = sinC / c 3) Choose the law and the formula that would be used to solve the triangle. Law: Law of Sines Formula: sinA / a = sinB / b 4) Choose the law and the formula that would be used to solve the triangle. Law: Law of Cosines Formula: a^2 = b^2 + c^2 - 2bcCOS A 5) Choose the law and the formula that would be used to solve the triangle. Law: Law of Cosines Formula: b^2 = a^2 + c^2 - 2acCOS B 6) Choose the law and the formula that would be used to solve the triangle. Law: Law of Cosines Formula: c^2 = a^2 + b^2 - 2abCOS C Find the length of side b . 2.85 cm Find the length of a . 3.7 cm Find m∠B to the nearest degree. 47° angle in standard position an angle with its vertex at the origin and one of its rays on the x-axis of the coordinate plane radian a unit of angular measure equal to the length of the arc divided by the radius of the arc reference angle the positive acute angle formed by the terminal side of an angle in standard position and the x-axis terminal side of an angle he ray of an angle in standard position that does not lie on the x-axis unit circle a circle with a radius of one unit that has its center at the origin on the coordinate plane