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5.2:Triangles and Right Triangle Trigonometry 1. 2. 3. 4. 5. Classifying Triangles Using the Pythagorean Theorem Understanding Similar Triangles Understanding Special Right Triangles Using Similar Triangles to Solve Applied Problems 6. Use right triangles to evaluate trigonometric functions. 7. Find function values for 8. Recognize and use fundamental identities. 9. Use equal cofunctions of complements. 10. Evaluate trigonometric functions with a calculator. 11. Use right triangle trigonometry to solve applied problem Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ [email protected] H.Melikian/1200 1 Classification of Triangles Triangles can be classified according to their angles: Acute: 3 acute angles Obtuse: One obtuse angle Right: One right angle Triangles can be classified according to their sides: Scalene: no congruent sides Isosceles: two congruent sides Equilateral: three congruent sides H.Melikian/1200 2 Classifying a Triangle Classify the given triangle as acute, obtuse, right, scalene, isosceles, or equilateral. State all that apply. The triangle is acute because all the angles are less than 90 degrees. The triangle is scalene since all the sides are different. The Pythagorean Theorem Given any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. h o o2 + a2 = h2. a H.Melikian/1200 3 Using the Pythagorean Theorem Use the Pythagorean Theorem to find the length of the missing side of the given right triangle. a b c 2 9 2 2 92 142 c2 14 81 196 c2 277 c 2 277 c what if 9 H.Melikian/1200 15 4 Similar Triangles Triangles that have the same shape but not necessarily the same size. 1. The corresponding angles have the same measure. 2. The ratio of the lengths of any two sides of one triangle is equal to the ratio of the lengths of the corresponding sides of the other triangle. Example: Triangles ABC and DEF are similar. Find the lengths of the missing sides of triangle ABC. A D B H.Melikian/1200 C E DE F EF AC AB DF DE AC BC DF EF 15 10 12 DE 15 8 12 EF DE 8 DE 8 5 H.Melikian/1200 6 Objectives: Use right triangles to evaluate trigonometric functions. , 45 , and 60 . 30 Find function values for 6 4 3 Recognize and use fundamental identities. Use equal cofunctions of complements. Evaluate trigonometric functions with a calculator. Use right triangle trigonometry to solve applied problems. H.Melikian/1200 7 The Six Trigonometric Functions The six trigonometric functions are: Function sine cosine tangent cosecant secant cotangent H.Melikian/1200 Abbreviation sin cos tan csc sec cot 8 Right Triangle Definitions of Trigonometric Functions In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle. H.Melikian/1200 9 Right Triangle Definitions of Trigonometric Functions(continued) In general, the trigonometric functions of depend only on the size of angle and not on the size of the triangle. H.Melikian/1200 10 Example: Evaluating Trigonometric Functions Find the value of the six trigonometric functions in the figure. We begin by finding c. a b c 2 2 2 c 2 32 42 9 16 25 c 25 5 3 sin 5 4 cos 5 H.Melikian/1200 3 tan 4 5 csc 3 5 sec 4 4 cot 3 11 Function Values for Some Special Angles A right triangle with a 45°, or 4 radian, angle is isosceles – that is, it has two sides of equal length. H.Melikian/1200 12 Function Values for Some Special Angles (continued) A right triangle that has a 30°, or radian, angle also has a 60°, or 3 6 radian angle. In a 30-60-90 triangle, the measure of the side opposite the 30° angle is onehalf the measure of the hypotenuse. H.Melikian/1200 13 Example: Evaluating Trigonometric Functions of 45° Use the figure to find csc 45°, sec 45°, and cot 45°. csc 45 sec 45 2 length of hypotenuse 1 length of side opposite 45 2 length of hypotenuse 2 length of side adjacent to 45 1 2 1 length of side adjacent to 45 1 cot 45 1 length of side opposite 45 H.Melikian/1200 14 Example: Evaluating Trigonometric Functions of 30°and 60° Use the figure to find tan 60° and tan 30°. If a radical appears in a denominator, rationalize the denominator. tan 60 length of side opposite 60 3 1 length of side adjacent to 60 tan 30 1 1 length of side opposite 30 3 3 length of side adjacent to 30 H.Melikian/1200 3 3 3 3 3 15 Trigonometric Functions of Special Angles H.Melikian/1200 16 Fundamental Identities H.Melikian/1200 17 Example: Using Quotient and Reciprocal Identities 2 and cos 5 find the value of each of the four Given sin 3 3 remaining trigonometric functions. sin tan cos csc H.Melikian/1200 2 2 3 5 3 3 3 2 2 5 5 5 5 2 5 5 5 1 1 3 2 sin 2 3 18 Example: Using Quotient and Reciprocal Identities (continued) sin 2 3 5 Given and cos 3 find the value of each of the four remaining trigonometric functions. sec 1 3 3 1 5 5 5 cos 5 3 5 5 5 3 5 1 1 5 cot 2 5 2 5 2 5 tan 5 5 5 5 25 2 5 5 H.Melikian/1200 19 The Pythagorean Identities H.Melikian/1200 20 Example: Using a Pythagorean Identity 1 sin Given that and is an acute angle, find the 2 value of cos using a trigonometric identity. sin 2 cos 2 1 2 1 cos 2 1 2 1 cos 2 1 4 cos 2 1 cos 2 cos H.Melikian/1200 1 4 3 4 3 3 4 2 21 Trigonometric Functions and Complements Two positive angles are complements if their sum is 90° or 2 . Any pair of trigonometric functions f and g for which f ( ) g (90 ) and g ( ) f (90 ) are called cofunctions. H.Melikian/1200 22 Cofunction Identities H.Melikian/1200 23 Using Cofunction Identities Find a cofunction with the same value as the given expression: a. sin 46 cos(90 46) cos 44 b. 6 5 tan cot tan tan 12 12 2 12 12 12 H.Melikian/1200 24 Using a Calculator to Evaluate Trigonometric Functions To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions. Example: Evaluating Trigonometric Functions with a Calculator Use a calculator to find the value to four decimal places: a. sin72.8° (hint: Be sure to set the calculator to degree mode) sin 72.8 0.9553 b. csc1.5 (hint: Be sure to set the calculator to radian mode) csc1.5 1.0025 H.Melikian/1200 25 Applications: Angle of Elevation and Angle of Depression An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. The angle formed by the horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression. H.Melikian/1200 26 Example: Problem Solving Using an Angle of Elevation The irregular blue shape in the figure represents a lake. The distance across the lake, a, is unknown. To find this distance, a surveyor took the measurements shown in the figure. What is the distance across the lake? tan 24 a 750 a 750 tan 24 a 333.9 The distance across the lake is approximately 333.9 yards. H.Melikian/1200 27