4-57.  To find out how high Juanisha climbed up stairs, you need to know more about the relationship between the ratios of the sides of a right triangle and the slope angle. Use two different strategies to determine Δy for the slope triangles shown in the diagram at right. 12 + b2 = 22 1 3 = 3 𝑥 Calculate the ratio Δy/hypotenuse for each triangle.  Why must these ratios be equal? 3 2 because the sides of similar triangles must be proportional Determine BC and AC in the triangle below.  Show all work. 1 2 = 𝐵𝐶 7 BC = 3.5 2 3 = 7 𝐴𝐶 AC = 3.5 3 AC = 6.06 3√3 Warm Up

HW: 4-62 through 4-67 4.2.1 Sine and Cosine Ratios November 20, 2015

Objectives: CO: SWBAT use sine and cosine ratios to solve for missing sides. LO: SWBAT explain when to use sine, cosine, or tangent.

4-58.  NEW TRIG RATIOS In problem 4-57, you used a ratio that included the hypotenuse of ΔABC.  There are several ratios that you might have used.  One of these ratios is known as the sine ratio (pronounced “sign”).  This is the ratio of the length of the side opposite the acute angle to the length of the hypotenuse. For the triangle shown at right, the sine of 60° is   ≈ 0.866.  This is written: sin 60° =  Another ratio comparing the length of the side adjacent to (which means “next to”) the angle to the length of the hypotenuse is called the cosine ratio (pronounced “co-sign”).  For the triangle above, the cosine of 60° is 1/2 = 0.5.  This is written: cos 60° = 1/2 1 2 Like the tangent ratio, your calculator can give you both the sine and cosine ratios for any angle.  Locate the “sin” and “cos” buttons on your calculator and use them to determine the sine and cosine of 60°.  Does your calculator give you the correct ratios? 4 3 team

Sine & Cosine are reciprocals 4-58.  NEW TRIG RATIOS Cosine = Adjacent/Hypotenuse Sine = Opposite/Hypotenuse Use a trig ratio to write an equation and solve for a in the diagram below.  Does this require the sine ratio or the cosine ratio? Sine sin(23) = a/15 15sin(23) = a 5.86 = a Likewise, write an equation and solve for b for the triangle below. Cosine cos(37) = b/8 8cos(37) = b 6.39 = b Together Sine & Cosine are reciprocals

4-60.  For each triangle below, decide which side is opposite and which is adjacent to the given acute angle.  Then determine which of the three trig ratios will help you solve for x.  Finally, write and solve an equation. 3 b. Sine sin(25) = x/9 9sin(25) = x 3.8 = x c. Tangent tan(45) = x/5 5tan(45) = x 5 = x a. Cosine cos(17) = x/3 3cos(17) = x 2.87 = x x x f. Cosine cos(20) = x/10 10cos(20) = x 9.4 = x e. Sine sin(34) = x/13 13sin(34) = x 7.27 = x ONE pencil (make them put the other one away!) A explains a to B. B writes in NB. B explains b to A. A writes in NB. Check with other pair. Continue to go back and forth. Check rest with other pair at end. x d. Tangent tan(62) = 5/x tan(28) = x/5 5tan(28) = x 2.66 = x 13 Pairs Check

4-61. TRIANGLE GRAPHIC ORGANIZER Think about the tools you have developed so far to solve for the measure of sides and angles of a triangle.  Then, in the spaces provided, add a diagram and a description of each tool you know. Pythagorean Theorem Tangent Sine Cosine