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Warm Up for Section 1.2 Simplify: (1). (2). (3). There are 10 boys and 12 girls in a Math 2 class. Write the ratio of the number of girls to the number of boys as a fraction in simplest form. (4). In a 30 o -60 o -90 o triangle, the hypotenuse measures 10 inches. What is the length of the long leg?

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Answers for Warm Up Section 1.2 Simplify: (1). (2). (3). There are 10 boys and 12 girls in a Math 2 class. Write the ratio of the number of girls to the number of boys as a fraction in simplest form. (4). In a 30 o -60 o -90 o triangle, the hypotenuse measures 10 inches. What is the length of the long leg?

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Work for Answers to WU, Section 1.2 (1). (2). (3). girls (4). boys 30 o 60 o

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Trigonometric Ratios Section 1.2 Standard: MM2G2 Essential Question: What is the definition of the sine, cosine, and tangent ratios?

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Investigation 1: (1). How do you find the missing side length? 10 8 x Use the Pythagorean Theorem to find the missing side length!

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(2). Does that work for this triangle? Why not? 31 o 5 To use the Pythagorean Theorem, we must know 2 of the 3 side lengths. We only know one side length, so this theorem cannot be used.

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We need something besides the Pythagorean Theorem to find missing measures for right triangles when we don’t know two side lengths. That something is Trigonometry! Trigonometry is a branch of mathematics that deals with the relationships between the sides of angles of triangles and the calculations based on these relationships. A trigonometric ratio describes a relationship between the sides and angles of a right triangle.

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(3). It is important to identify the correct side when creating a trig ratio. Let’s investigate ABC below and its sides: a, b, and c. With your partner, identify each of the following: hypotenuse: _______ side opposite angle A: _______ side adjacent to angle A: _______ side opposite angle B: _______ side adjacent to angle B: _______ A B C b a c c a b b a

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Recall: Similar triangles are triangles that have the same shape but not necessarily the same size. (4). In similar triangles, corresponding angles are congruent and corresponding sides are proportional. Consider the four similar triangles pictured below. How do you know the triangles are similar? Ans: Because corresponding angles are congruent

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For the 31 o angle in each triangle, write the ratio of : 3 5 9 15 6 10 33 55 = = =

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Now, find the missing side, x, by setting up a proportion and solving? Then, how could you find y? 31 o 8 x y Once we know x, we can use it and 8 in the Pythagorean Theorem to find y.

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Early mathematicians found these ratios of similar triangles very useful and they named them the Trigonometric Ratios. They made tables of these calculations. Today, we can obtain these values with a calculator! The ratios we will use are listed below – memorize them!

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tangent of θ = Length of leg opposite θ Length of leg adjacent to θ Length of leg opposite θ Length of hypotenuse Length of leg adjacent to θ Length of hypotenuse For any acute angle in a right triangle, we denote the measure of the angle by θ and define three ratios related to θ as follows: sine of θ = cosine of θ =

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In the figure below, the terms “opposite,” “adjacent,” and “hypotenuse” are used as shorthand for the lengths of these sides. Using this shorthand, we can give abbreviated versions of the above definitions: tan = sin = cos = opposite adjacent opposite hypotenuse adjacent hypotenuse SOH-CAH-TOA The Indian princess will help you remember !!

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Check for Understanding: Write each trigonometric ratio for the right triangle pictured below: sin A = ____=___ tan A = ____=___ cos B = ____=___ cos A = ___=____ sin B = ___=____ tan B = ___=___ 15 12 9 A B C 15 12 9 12 15 9 15 9 15 9 12 4545 4545 4343 3434 3535 3535

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(5). sin P = (6). sin S = (7). sin L = (8). cos P = (9). cos T = (10). cos M = (12). tan P = (13). tan S = (14). tan L = 13 T 12 5 S U 10 8 P Q R 6 17 15 L N M 8 6 10 8 10 6868 5 13 5 13 5 12 8 17 8 17 8 15

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Investigation 2: Use your calculator to estimate each trig ratio to the nearest thousandth (three digits after the decimal). Follow the steps below to estimate each value. Step 1: Set you calculator to degree mode. Step 2: Press the desired ratio (tan, sin, cos), type the angle measure in degrees, press ENTER. Step 3: Round your answer to three digits after the decimal.

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(14). cos(47 o ) ____________ (15). sin(63 o ) _____________ (16). tan(12 o ) ______________ 0.682 0.891 0.213

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Investiagion 3: Write and solve an equation using a trig ratio for 48 o using 10 and x. Use your calculator to estimate the value of the trig ratio to three digits after the decimal. Round your answer to three digits as well. tan (48 o ) = x 10 10 (tan 48 o ) = x 11.106 x

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Check for Understanding: For each diagram, write a trig ratio for the given acute angle using x and the given side. Solve for x. Round answers to the nearest thousandth. (17). x 20 50° cos (50 o ) = 20 x x (cos 50 o ) = 20 x = x 31.114 20 cos 50 o

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20° x 12 (18). sin (20 o ) = 12 (sin 20 o ) = x x 12 4.104 x

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(19). 12 x 25° tan(25 o ) = x = 12 x x ≈ 25.7

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Formula Sheet: Trigonometric Ratios: SOH-CAH-TOA sin θ = cos θ = tan θ = θ Opp Adj Hyp

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