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H.Melikian/12001 5.2:Triangles and Right Triangle Trigonometry Dr.Hayk Melikyan/ Departmen of Mathematics and CS/ melikyan@nccu.edu 1. Classifying Triangles 2. Using the Pythagorean Theorem 3. Understanding Similar Triangles 4. Understanding Special Right Triangles 5. 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

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H.Melikian/12002 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

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H.Melikian/12003 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. o a h o 2 + a 2 = h 2.

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H.Melikian/12004 Using the Pythagorean Theorem Use the Pythagorean Theorem to find the length of the missing side of the given right triangle. 9 14 what if 9 15

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H.Melikian/12005 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.

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H.Melikian/12006

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7 Objectives: Use right triangles to evaluate trigonometric functions. Find function values for Recognize and use fundamental identities. Use equal cofunctions of complements. Evaluate trigonometric functions with a calculator. Use right triangle trigonometry to solve applied problems.

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H.Melikian/12008 The Six Trigonometric Functions The six trigonometric functions are: FunctionAbbreviation sine sin cosine cos tangent tan cosecant csc secant sec cotangent cot

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H.Melikian/12009 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.

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H.Melikian/120010 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.

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H.Melikian/120011 Example: Evaluating Trigonometric Functions Find the value of the six trigonometric functions in the figure. We begin by finding c.

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H.Melikian/120012 Function Values for Some Special Angles A right triangle with a 45°, or radian, angle is isosceles – that is, it has two sides of equal length.

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H.Melikian/120013 Function Values for Some Special Angles (continued) A right triangle that has a 30°, or radian, angle also has a 60°, or radian angle. In a 30-60-90 triangle, the measure of the side opposite the 30° angle is one- half the measure of the hypotenuse.

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H.Melikian/120014 Example: Evaluating Trigonometric Functions of 45° Use the figure to find csc 45°, sec 45°, and cot 45°.

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H.Melikian/120015 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.

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H.Melikian/120016 Trigonometric Functions of Special Angles

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H.Melikian/120017 Fundamental Identities

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H.Melikian/120018 Example: Using Quotient and Reciprocal Identities Given and find the value of each of the four remaining trigonometric functions.

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H.Melikian/120019 Example: Using Quotient and Reciprocal Identities (continued) v Given and find the value of each of the four remaining trigonometric functions.

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H.Melikian/120020 The Pythagorean Identities

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H.Melikian/120021 Example: Using a Pythagorean Identity Given that and is an acute angle, find the value of using a trigonometric identity.

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H.Melikian/120022 Trigonometric Functions and Complements Two positive angles are complements if their sum is 90° or Any pair of trigonometric functions f and g for which and are called cofunctions. v

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H.Melikian/120023 Cofunction Identities

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H.Melikian/120024 Using Cofunction Identities Find a cofunction with the same value as the given expression: a. b.

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H.Melikian/120025 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. Use a calculator to find the value to four decimal places: a. sin72.8° (hint: Be sure to set the calculator to degree mode) b. csc1.5 (hint: Be sure to set the calculator to radian mode) Example: Evaluating Trigonometric Functions with a Calculator

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H.Melikian/120026 Applications: Angle of Elevation and Angle of Depression v 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.

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H.Melikian/120027 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? The distance across the lake is approximately 333.9 yards.

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