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Trigonometric Ratios A RATIO is a comparison of two numbers. For example; boys to girls cats : dogs right : wrong. In Trigonometry, the comparison is between sides of a triangle ( right triangle).

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Warm up Solve the equations: A) 0.875 = x/18 B) 24/y =.5 C) y/25 =.96

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E.Q: How can we find the sin, cosine, and the tangent of an acute angle? How do we use trigonometric ratios to solve real-life problems?

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Trig. Ratios Name “say” SineCosinetangent Abbreviation Abbrev. SinCosTan Ratio of an angle measure Sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side

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Easy way to remember trig ratios: SOH CAH TOA Three Trigonometric Ratios Sine – abbreviated ‘sin’. –Ratio: sin θ = opposite side hypotenuse Cosine - abbreviated ‘cos’. –Ratio: cos θ = adjacent side hypotenuse Tangent - abbreviated ‘tan’. –Ratio: tan θ = opposite side adjacent side Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.

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Let’s practice… B c a C b A Write the ratio for sin A Sin A = o = a h c Write the ratio for cos A Cos A = a = b h c Write the ratio for tan A Tan A = o = a a b Let’s switch angles: Find the sin, cos and tan for Angle B: Sin B = b c Cos B = a c Tan B = b a

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Make sure you have a calculator… I want to findUse these calculator keys sin, cos or tan ratio SIN COS TAN Angle measure SIN -1 COS -1 TAN -1 Set your calculator to ‘Degree’….. MODE (next to 2 nd button) Degree (third line down… highlight it) 2 nd Quit

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Let’s practice… C 2cm B 3cm A Find an angle that has a tangent (ratio) of 2 3 Round your answer to the nearest degree. Process: I want to find an ANGLE I was given the sides (ratio) Tangent is opp adj TAN -1 (2/3) = 34°

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Practice some more… Find tan A: 24.19 12 A 21 Tan A = opp/adj = 12/21 Tan A =.5714 8 4 A Tan A = 8/4 = 2 8 Find tan A:

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Trigonometric Ratios When do we use them? –On right triangles that are NOT 45-45-90 or 30-60-90 Find: tan 45 1 Why? tan = opp hyp

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Using trig ratios in equations Remember back in 1 st grade when you had to solve: 12 = x What did you do? 6 (6) 72 = x Remember back in 3rd grade when x was in the denominator? 12 = 6 What did you do? x (x) 12x = 6 __ 12 x = 1/2

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x cm 15 cm 34° Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse Ask yourself: What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: Sin 34 = x 15 (15) (15)Sin 34 = x 8.39 cm = x

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x cm 12 cm 53° Ask yourself: In relation to the angle, what pieces do I have? Opposite and adjacent Ask yourself: What trig ratio uses Opposite and adjacent? tangent Set up the equation and solve: Tan 53 = x 12 (12) (12)tan 53 = x 15.92 cm = x

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x cm 18 cm 68° Ask yourself: In relation to the angle, what pieces do I have? Adjacent and hypotenuse Ask yourself: What trig ratio uses adjacent and hypotnuse? cosine Set up the equation and solve: Cos 68 = 18 x (x) (x)Cos 68 = 18 X = 18 cos 68 _____ cos 68 X = 48.05 cm

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42 cm 22 cm θ This time, you’re looking for theta. Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse Ask yourself: What trig ratio uses opposite and hypotenuse? sine Set up the equation (remember you’re looking for theta): Sin θ = 22 42 Remember to use the inverse function when you find theta THIS IS IMPORTANT!! Sin -1 22 = θ 42 31.59°= θ

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17 cm 22 cm θ You’re still looking for theta. Ask yourself: What trig ratio uses the parts I was given? tangent Set it up, solve it, tell me what you get. tan θ = 17 22 THIS IS IMPORTANT!! tan -1 17 = θ 22 37.69°= θ

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Using trig ratios in equations Remember back in 1 st grade when you had to solve: 12 = x What did you do? 6 (6) 72 = x Remember back in 3rd grade when x was in the denominator? 12 = 6 What did you do? x (x) 12x = 6 __ 12 x = 1/2

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Types of Angles The angle that your line of sight makes with a line drawn horizontally. Angle of Elevation Angle of Depression

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SOA CAH TOA

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