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Right Triangle Trigonometry Section 13.1 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 1 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° 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 2 Right Triangles hypotenuse opposite adjacent • 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 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 3 Right Triangles SOHCAHTOA opposite Sine ө= hypotenuse SIN Cosine ө = COS opposite adjacent Tangent ө = adjacent hypotenuse TAN Reciprocals of SOHCAHTOA adjacent hypotenuse hypotenuse Cosecant ө = Secant ө = Cotangent ө= adjacent opposite opposite CSC 4/6/2016 5:26 PM SEC 13.1 Right Triangle Trigonometry COT 4 Right Triangles Relationships of Trigonometric Ratios Sine ө = SIN opposite hypotenuse Cosine ө = COS Cosecant ө = CSC adjacent hypotenuse Tangent ө = TAN 4/6/2016 5:26 PM opposite adjacent Secant ө = SEC hypotenuse adjacent Cotangent ө = COT 13.1 Right Triangle Trigonometry hypotenuse opposite adjacent opposite 5 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 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 6 Example 1 Find x and determine all trig functions of ө 13 Step 1: Find x 12 x Use the Pythagorean Theorem to find the length of the adjacent side… a2 + 122 = 132 a2 = 25 a=5 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 7 Example 1 Find x and determine all trig functions of ө 13 12 x Step 2: Determine the hypotenuse and the opposite by identifying ө adj = 5 4/6/2016 5:26 PM opp = 12 13.1 Right Triangle Trigonometry hyp = 13 8 Example 1 Find x and determine all trig functions of ө 13 12 5 Step 3: Use Trigonometry Functions to find what’s needed Sine ө= SIN opposite hypotenuse Cosine ө = COS adjacent Tangent ө = opposite adjacent hypotenuse TAN Cosecant ө =hypotenuse Secant ө =hypotenuse Cotangent ө= CSC 4/6/2016 5:26 PM opposite SEC adjacent 13.1 Right Triangle Trigonometry COT adjacent opposite 9 Example 1 Find x and determine all trig functions of ө 13 12 5 Step 3: Use Trigonometry Functions to find what’s needed Sine ө= SIN 12 13 Cosecant ө = CSC 4/6/2016 5:26 PM Cosine ө = COS 13 12 Secant ө = SEC 5 13 13 5 13.1 Right Triangle Trigonometry Tangent ө = TAN Cotangent ө= COT 12 5 5 12 10 Your Turn Determine all trig functions of ө 2 4/6/2016 5:26 PM 3 1 13.1 Right Triangle Trigonometry 11 Your Turn Determine all trig functions of ө 2 Sine ө= SIN Cosecant ө = CSC 3 2 3 1 Cosine ө = COS 2 3 Secant ө = SEC 1 2 2 Tangent ө = 3 TAN Cotangent ө= COT 1 3 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 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 12 Example 2 What is given? – Hypotenuse: 74 – Opposite of 30°: x – Adjacent: Unknown Solve for x. Which of the six trig ratios is best fit for this triangle? (there can be more than one answer) 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 13 Example 2 Which of the six trig ratios is best fit for this triangle? (there can be more than one answer) Solve for x. opposite sin hypotenuse opposite sin 30 hypotenuse x sin 30 Must change 74 the answer to DEGREE x 74sin 30 mode and not RADIAN x 37 mode in calculator 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 14 Example a Find the value of sine, cosine and tangent functions 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? opposite Substitute 15.1° for θ, h for opposite, hypotenuse and 19 for hypotenuse. h sin15.1 Multiply both sides by 19. 19 Use a calculator to simplify. h 4.9496 sin The height above the water is about 5 ft. 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 16 Your Turn Solve for h. Round to 4 decimal places 0.6765 km 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 17 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 B 55 a 8.9879 4/6/2016 5:26 PM b 12.8361 13.1 Right Triangle Trigonometry C 90 18 Example b Find the value of x 10 Example c Find the value of x. 12 Example d Find the value of x. 8 Angle of Elevation vs. Depression • Angle of Elevation is a measurement above the horizontal line • Angle of Depression is a measurement below the horizontal line Angle of Elevation Angle of Depression 4/6/2016 5:26 PM 6.2 Trig Applications 22 Example 4 A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole. 35° ---- 60 Feet ---4/6/2016 5:26 PM 6.2 Trig Applications 23 Example 4 A flagpole casts a 60-foot shadow when the angle of elevation of the sun is 35°. Find the height of the flagpole. x tan 35 60 35° ---- 60 Feet ---- x 42.0125 ft. 4/6/2016 5:26 PM 6.2 Trig Applications 24 Example 5 Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°. 6 100 ft. 6 ? 4/6/2016 5:26 PM 951.4364 ft. 6.2 Trig Applications 25 Your Turn 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. 4/6/2016 5:26 PM 17.9513 ft. 6.2 Trig Applications 26 Example e Solve 45 ft Assignment Pg 933 3-25 odd 4/6/2016 5:26 PM 13.1 Right Triangle Trigonometry 28