Essential Question? How can we use triangles, especially right triangles, to solve problems?

Properties of Rational Exponents PropertyExample 1. a m ∙ a n = a m+n 2. (a m ) n = a mn 3. (ab) m = a m b m

Warm up Inverse Variation: Boyle’s Law states that when a sample of gas is kept at a constant temperature, the volume, V varies inversely with the pressure, P exerted on it. Write an equation for Boyle’s Law If V = 20 Liters at 500 psi, what is V if pressures is 800 psi

Trigonometric Ratios A RATIO is a comparison of two numbers. For example; boys to girls cats : dogs right : wrong. Trigonometry – study of the measurement of sides and angles in triangles In Trigonometry, the comparison is between sides of a right triangle.

Three Trigonometric Ratios Sine – abbreviated ‘sin’. Ratio: sin θ = opposite side hypotenuse Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’. A C B opposite hypotenuse θ

Three Trigonometric Ratios Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’. A C B Cosine - abbreviated ‘cos’. Ratio: cos θ = adjacent side hypotenuse adjacent hypotenuse θ

Three Trigonometric Ratios Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’. A C B Tangent - abbreviated ‘tan’. Ratio: tan θ = opposite side adjacent side opposite adjacent θ

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’.

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

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 To set your calculator to ‘Degree’….. Press MODE (next to 2 nd button) Degree (third line down… highlight it by pressing Enter 2 nd Quit Clear

Let’s practice… B c a C b A Sin Θ = Opposite Hypotenuse Cos Θ = Adjacent Hypotenuse Tan Θ = Opposite Adjacent Sin A=Sin B = Cos A=Cos B = Tan A=Tan B =

Lesson 4.4 (I)

Ex 1) How do we find the angle measure? CA B ΘoΘo 18 cm 12 2 nd Cos(12/18) = Cos -1 (12/18) = 48.2 o 1) What is given? 2) What trig ratio? 3) What is asked for? Find measure of <B? Hypotenuse Adjacent Cos Θ = adj/hyp Find angle Θ =

Using trig ratios in equations Remember when you had to solve: 12 = x What did you do? 6 (6) 72 = x What if x is in the denominator? 12 = 6 What did you do? x (x) 12x = 6 __ 12 x = 1/2

Ex 2) Let’s practice… B C A Process: 1)Identify what is given 2)Which trig ratio, sin, cos, or tan will work with what is given 3)Plug in and solve X cm 40 o 7.6 cm Process: 1)Hyp = 7.6 <A = 40 o and opposite = x 2) Sin = opposite/hypotenuse 3) solve: 7.6 cm X cm Sin 40 o = 7.6 x Sin 40 o X = X = 4.9 cm

Ex 3) Let’s practice… B c A Process: 1)Identify what is given 2)Which trig ratio, sin, cos, or tan will work with what is given 3)Plug in and solve X cm 36 o 18 cm Process: 1)Hyp = 18 <B = 36 o and adjacent = x 2) Cos = adjacent/hypotenuse 3) Solve: 18 cm X cm Cos 36 o = 18 x Cos 36 o X = X = 14.6 cm

Ex 4) Let’s practice c A X cm 30 o 18 cm Process: 1)Hyp = x <A = 30 o and adjacent = 18 2) Cos = adjacent/hypotenuse 3) Solve: 18 cm X cm Cos 30 o = 18 cm X cm Cos30 o and x have to change places – Swith and divide! X cm Cos 30 o = 18 cm Cos 30 o X cm = 18 cm Cos 30 o = 20.8 cm B

Practice some more… Ex 5) Find tan A: CA B 48 o 5.8 x C A B 54 o Ex 6) What trig function would find x? 18 x

Toolkit Trig Ratios Unknown will be in one of three places: Sin Θ =Angle Θ Cos Θ =Numerator: Tan Θ =Denominator: Opposite Hypotenuse Adjacent Hypotenuse Opposite Adjacent 2 nd trig(ratio) = angle x given Trig angle = given x Trig angle = Multiply Switch and divide

Warm up Google 1)When and where did Pythagoras live? 2)How old is the Great Pyramid of Egypt? 3)Is it an equilateral triangle? What is the base length?

Quiz Draw and label each triangle and find what is asked for below: 1.Let side c = 15 ft. and side b = 9 ft. Find angle A and side a A B C c a b