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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry1 Right Triangle Trigonometry Section 13.1

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry2 Definitions Trigonometry –Comes from Greek word – Trigonon, which means 3 angles –“Metry” means measure in Greek Trigonometry Ratios –Sine, Cosine, Tangent, Secant, Cosecant, Cotangent Types of angles –Acute: Less than 90° –Equilateral: 90° –Obtuse: More than 90° but less than 180°

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry3 Right Triangles Consider a right triangle, one of whose acute angles is ө The three sides of a triangle are hypotenuse, opposite, and adjacent side of ө To determine what is the opposite side, look at ө and extend the line to determine the opposite hypotenuse opposite adjacent

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry4 Right Triangles SOHCAHTOA Sine ө= Cosine ө = Tangent ө = SIN COS TAN Reciprocals of SOHCAHTOA Cosecant ө = Secant ө = Cotangent ө= CSC SEC COT

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry5 Relationships of Trigonometric Ratios Sine ө = Cosecant ө = SINCSC Cosine ө =Secant ө = COSSEC Tangent ө = Cotangent ө = TANCOT Right Triangles

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry6 Steps in Determining Triangles 1.Solve for x, using Pythagorean Theorem 2.Determine the hypotenuse and the opposite by identifying ө 3.Use Trigonometry Functions to find what’s needed

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry7 Example 1 Step 1: Find x 13 12 x Find x and determine all trig functions of ө Use the Pythagorean Theorem to find the length of the adjacent side… a 2 + 12 2 = 13 2 a 2 = 25 a = 5

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry8 Example 1 Step 2: Determine the hypotenuse and the opposite by identifying ө 13 12 x Find x and determine all trig functions of ө adj = 5opp = 12hyp = 13

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry9 Example 1 Step 3: Use Trigonometry Functions to find what’s needed Find x and determine all trig functions of ө Sine ө= Cosine ө = Tangent ө = SIN COS TAN Cosecant ө = Secant ө = Cotangent ө= CSC SEC COT 13131212 55

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry10 Example 1 Find x and determine all trig functions of ө 13 12 55 Sine ө= Cosine ө = Tangent ө = SIN COS TAN Cosecant ө = Secant ө = Cotangent ө= CSC SEC COT Step 3: Use Trigonometry Functions to find what’s needed

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry11 Your Turn 22 11 Determine all trig functions of ө

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry12 Your Turn Determine all trig functions of ө 22 11 Sine ө= Cosine ө = Tangent ө = SIN COS TAN Cosecant ө = Secant ө = Cotangent ө= CSC SEC COT Can we have radicals in the denominators? Actually, with trig ratios, it is accepted in the subject area. But it is necessary to simplify radicals

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry13 Example 2 What is given? –Hypotenuse: 74 –Opposite of 30°: x –Adjacent: Unknown Which of the six trig ratios is best fit for this triangle? (there can be more than one answer) Solve for x.

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry14 Example 2 Which of the six trig ratios is best fit for this triangle? (there can be more than one answer) Solve for x. Must change the answer to DEGREE mode and not RADIAN mode in calculator

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Example a Find the value of sine, cosine and tangent functions

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry16 Example 3 In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp? The height above the water is about 5 ft. Substitute 15.1° for θ, h for opposite, and 19 for hypotenuse. Multiply both sides by 19. Use a calculator to simplify.

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry17 Your Turn Solve for h. Round to 4 decimal places

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry18 Your Turn Solve for the rest of missing sides of triangle ABC, given that A = 35° and c = 15.67. Round to 4 decimal places

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Example b Find the value of x 10

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Example c Find the value of x. 12

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Example d Find the value of x. 8

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6/10/2015 8:06 AM6.2 Trig Applications22 Angle of Elevation is a measurement above the horizontal lineAngle of Elevation is a measurement above the horizontal line Angle of Depression is a measurement below the horizontal lineAngle of Depression is a measurement below the horizontal line Angle of Elevation Angle of Depression Angle of Elevation vs. Depression

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6/10/2015 8:06 AM6.2 Trig Applications23 A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole. Example 4 35° ---- 60 Feet ----

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6/10/2015 8:06 AM6.2 Trig Applications24 A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole. Example 4 35° ---- 60 Feet ----

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6/10/2015 8:06 AM6.2 Trig Applications25 Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°. Example 5

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6/10/2015 8:06 AM6.2 Trig Applications26 A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree. Your Turn

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Example e Solve 45 ft

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6/10/2015 8:06 AM13.1 Right Triangle Trigonometry28 Assignment Pg 933 3-25 odd

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