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1 Right Triangle Trigonometry.

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opposite hypotenuse adjacent hypotenuse adjacent opposite reference angle Anatomy of a Right Triangle

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Right Triangle Trigonometry3 If the reference angle is then Opp Hyp Adj then Opp Adj Hyp

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The Trigonometric Functions we will be looking at SINE COSINE TANGENT

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The Trigonometric Functions SINE COSINE TANGENT

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SINE Prounounced “sign”

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Prounounced “co-sign” COSINE

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Prounounced “tan-gent” TANGENT

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Prounounced “theta” Greek Letter Represents an unknown angle Sometimes called the reference angle or angle of perspective

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opposite hypotenuse adjacent hypotenuse opposite adjacent Definitions of Trig Ratios

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We need a way to remember all of these ratios…

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SOHCAHTOA Old Hippie Sin Opp Hyp Cos Adj Hyp Tan Opp Adj

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Some Old Hippie Came A Hoppin’ Through Our Apartment

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Finding sin, cos, and tan

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6 8 10 SOHCAHTOA

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Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 9 6 10.8 A

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Find the values of the three trigonometric functions of . 4 3 ? Pythagorean Theorem: (3)² + (4)² = c² 5 = c 5

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Find the sine, the cosine, and the tangent of angle A A 24.5 23.1 8.2 B Give a fraction and decimal answer (round to 4 decimal places).

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Finding a side

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Right Triangle Trigonometry20 Example: Find the value of x. Step 1: Identify the “reference angle”. Step 2: Label the sides (Hyp / Opp / Adj). Step 3: Select a trigonometry ratio (sin/ cos / tan). Sin = Step 4: Substitute the values into the equation. Sin 25 = Step 5: Solve the equation. reference angle Hyp opp Adj x = 12 sin 25 x = 12 (.4226) x = 5.07 cm =

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21 Solving Trigonometric Equations sin = sin 25 = x = (12) (sin 25 ) x = 5.04 cm sin 25 = x = x = 28.4 cm Sin = sin x = x = sin (12/25) x = 28.7 There are only three possibilities for the placement of the variable ‘x”. Note you are looking for an angle here!

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Angle of Elevation and Depression Example #1

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Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7 o. If the lighthouse is 150 ft tall, how far away is the sailboat? Construct a triangle and label the known parts. Use a variable for the unknown value. 5.7 o 150 ft. x

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Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7 o. If the lighthouse is 150 ft tall, how far away is the sailboat? 5.7 o 150 ft. x Set up an equation and solve.

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Angle of Elevation and Depression 5.7 o 150 ft. x Remember to use degree mode! x is approximately 1,503 ft.

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Angle of Elevation and Depression Example #2

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Angle of Elevation and Depression A spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35 o. The angle of elevation to the top of the spire is 38 o. How tall is the spire? Construct the required triangles and label. 500 ft. 38 o 35 o

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Angle of Elevation and Depression Write an equation and solve. Total height (t) = building height (b) + spire height (s) 500 ft. 38 o 35 o Solve for the spire height. t b s Total Height

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Angle of Elevation and Depression Write an equation and solve. 500 ft. 38 o 35 o Building Height t b s

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Angle of Elevation and Depression Write an equation and solve. 500 ft. 38 o 35 o The height of the spire is approximately 41 feet. t b s Total height (t) = building height (b) + spire height (s)

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Angle of Elevation and Depression Example #3

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Angle of Elevation and Depression A hiker measures the angle of elevation to a mountain peak in the distance at 28 o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29 o. How much higher is the mountain peak than the hiker? Construct a diagram and label. 1 st measurement 28 o. 2 nd measurement 1,500 ft closer is 29 o.

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Angle of Elevation and Depression Adding labels to the diagram, we need to find h. 28 o 29 o 1500 ftx ft h ft Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28 o is 1500 + x.

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Angle of Elevation and Depression Now we have two equations with two variables. Solve by substitution. Solve each equation for h. Substitute.

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Angle of Elevation and Depression Solve for x. Distribute. Get the x’s on one side and factor out the x. Divide. x = 35,291 ft.

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Angle of Elevation and Depression However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height. x = 35,291 ft. The height of the mountain above the hiker is 19,562 ft.

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A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? 50 71.5° ? tan 71.5° y = 50 (tan 71.5°) y = 50 (2.98868) Ex. 4

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A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? 200 x Ex. 5 60° cos 60° x (cos 60°) = 200 x X = 400 yards

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