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2 2 PRE-CALCULUS 11

3 Part 1: Trigonometry (10 marks) Triangle ABC with reference angle can be labelled with the following: B c = hypotenuse a = opposite A θ b = adjacent C The following table summarizes the three trigonometric ratios that you learned in a previous math course. Ratio Name Sine Cosine Description Calculation Mnemonic* The ratio of the length of the side opposite the reference angle to the length of the hypotenuse. The ratio of the length of the side adjacent to the reference angle to the length of the hypotenuse. Tangent The ratio of the length of the side opposite the reference angle to the length of the side adjacent to the reference angle. *A way to remember sinθ = cosθ = opposite hypotenuse adjacent hypotenuse tanθ = opposite adjacent S O H C A H T O A PRE-CALCULUS 11 3

4 Example 1 Determine the measure of length x to the nearest tenth. x Solution Label the triangle. 200 adjacent 43 x opposite 200 Given: one angle and the adjacent side Need: the opposite side Solve with tangent. SOH CAH TOA tan! = opposite adjacent tan 43 = x 200! ( 200) ( tan 43 ) = x \$ "# 200 %& ( 200) ( 200)tan( 43 ) = x x = PRE-CALCULUS 11

5 INTRoDUCTIoN ASSIgNMENT Example 2 Determine the measure of angle A to the nearest degree. B 8 cm 4 cm A C Solution Label the triangle. hypotenuse 8 4 opposite Given: the side opposite and the hypotenuse Need: angle A Solve with sine. SOH CAH TOA A = sin 1 opposite sin A = hypotenuse 4 sin A = 8 sin A = 0.5 ( sin A)= sin 11 ( 05. ) A = 30 To undo sin A and discover the measure of A, find the sin -1 of both sides of the equation. Solving a triangle involves finding all the remaining measurements of a triangle when you re given the measure of one length and one acute angle, or the measures of two lengths. Two things to remember: the sum of the interior angles of a triangle is 180, so if you re given the measure of one angle besides the right angle, you can calculate the measure of the third angle using addition and subtraction. you can solve for the third side of a right triangle using the Pythagorean Theorem, a 2 + b 2 = c 2. PRE-CALCULUS 11 5

6 Marks Now it s your turn. 1. Label the following triangle PQR with respect to the reference angle, x, where the right angle is at Q (there are two different ways to label it; choose one). Include the following labels: P, Q, R, p, q, r, opposite, adjacent, hypotenuse. (3 marks) x 6 PRE-CALCULUS 11

7 Marks 2. Solve the following right triangles. Answer to the nearest tenth. Include units with your answers. (7 marks; 4 marks for a and 3 marks for b) a. A 9.2 cm A = B = B 9.0 cm C AB = b. R R = RN = MR = M ft N PRE-CALCULUS 11 7

8 Part 2: Factoring Trinomials (10 marks) You can factor trinomials of the form ax 2 + bx + c by decomposition, using the tic-tac-toe method, or by inspection. Following are two examples showing the decomposition and tic-tac-toe methods. Example 1 Factor 2x 2 + 7x 15 by decomposition. Solution Find two numbers whose product is equal to the product of (2)( 15), which is 30, and whose sum is equal to +7, the middle or b term. By using a table or doing it in your head, you find that those numbers are +10 and 3. (+10)( 3) = ( 3) = +7 Rewrite the trinomial as a four-term polynomial using the two numbers just found as the coefficients of the middle two terms. 2x x 3x 15 Group the first two and the last two terms. (2x x) + ( 3x 15) Factor out the common factor in each group. 2x(x + 5) 3(x + 5) Factor out the common binomial. (2x 3) (x + 5) 8 PRE-CALCULUS 11

9 Example 2 Factor 3x 2 8x + 4 using the tic-tac-toe method. Solution Find two numbers whose product is (3)(4), which is 12, and whose sum is 8. These numbers are 6 and 2. Construct a 3 3 array as shown. Place the x 2 -term of the trinomial in the centre square and the constant in the last square in the bottom row. 3x 2 +4 Place an x-term in the last square in the middle row, and an x-term in the second square of the last row. These x-terms should have coefficients equal to the numbers determined earlier. 3x 2 6x 2x +4 Now find the Greatest Common Factor (GCF) of each row and column and write them in the row and column headings. For now, make them all positive. GCF of 3x 2 and 2x is x. GCF of 6x and 4 is 2. x 2 GCF of 3x 2 and 6x is 3x. 3x 3x 2 6x GCF of 2x and 3 is x +4 Now we ll check the sign of each GCF. 3x 2 is positive, so x and 3x can stay positive. However, the product of 3x and the other factor (currently +2) PRE-CALCULUS 11 9

10 must be 6x, so we ll make the 2 negative. In the same way, the product of x and the other factor must be 2x, so we ll make the other 2 negative as well. Our final chart looks like this: x 2 3x 3x 2 6x 2 2x +4 Now you can read the two binomial factors from the top row and the leftmost column of the grid. (x 2)(3x 2) = 3x 2 8x + 4 When factoring trinomials, you may also see a special case called the difference of squares. It s like a trinomial where the middle term has a coefficient of zero. Here are some examples. x x m 2 81 To factor a difference of squares, take the square root of each term. Then write one factor as the sum of the square roots and the other factor as the difference of the square roots. Example 3 Factor 9x 2 4y 2 completely and check by multiplying. Solution Step 1: First, check the terms for common factors. Since there are none, continue by determining the square roots of 9x 2 and 4y x = 3x 2 4y = 2y Step 2: Write one factor as the sum of the square roots. Write the other factors as the difference of the square roots. 9x 2 4y 2 = ( 3x+ 2y) ( 3x 2y) 10 PRE-CALCULUS 11

11 Marks Step 3: Check by multiplying. ( 3x+ 2y) ( 3x 2y) = 3x( 3x 2y)+ 2y( 3x 2y) 2 = 9x 6xy+ 6xy 4y = 9x 4y The original difference of squares is recovered, so the factors are correct. As the example above suggests, before you start factoring by decomposition or using the tic-tac-toe method, check for a common factor in each term. Now it s your turn. 1. Factor the following. (10 marks; 1 mark each) x 2 + x 6 2x x x 2 7x + 2 2x 2 + 7x + 6 6x x 10 x 2 + 6x + 9 PRE-CALCULUS 11 11

12 Marks 2x 2 13x + 6 x 2 6x + 8 4x x x 2 144y 2 12 PRE-CALCULUS 11

14 Marks b. y = ½(x + 2) 2 1 x y 14 PRE-CALCULUS 11

15 Marks 2. Using the tables of values from above, sketch both of the graphs. (5 marks) Note the following: The graphs are continuous, so join the points with lines. The graphs continue past the points we re graphing, so put arrows on the ends of the lines. a. y = 2(x 1) y x PRE-CALCULUS 11 15

16 Marks b. y = ½(x + 2) 2 1 y x 3. Analyze the graphs by answering the following. Write at least two points for each question. (4 marks) a. What is similar in both graphs? b. What is different about the graphs? 16 PRE-CALCULUS 11

17 Marks 4. Each graph above has a vertex, the lowest or highest point on the graph (depending on whether it opens upward or opens downward ). What is the vertex for each graph? (2 marks) y = 2(x 1) y = ½(x + 2) 2 1 PRE-CALCULUS 11 17

19 Example 1 Write 2 5 as an entire radical. Solution We need to write the 2 as a number under a radical sign. Method 1: Perfect Square Factors The number 2 can be written as 2 2 or 4. Then 2 5 = 4 5 = 20 Method 2: Prime Factorization Remember that the coefficient of 2 came from a pair of 2s under the radical symbol. Write 2 as 2 2. Then 2 5 = = 20 Example 2 3 Write 4 5 as an entire radical. Solution Using the method of prime factorization, write 4 as or 64 Then = = 64 5 = PRE-CALCULUS 11 19

20 Marks Now it s your turn. 1. Convert the following entire radicals to mixed radicals (simplify). (5 marks; ½ mark each) a. 20 b. 45 c. 72 d e f g. 150 h i. 112 j PRE-CALCULUS 11

21 Marks 2. Convert the following mixed radicals to entire radicals. (5 marks; ½ mark each) a. 2 3 b. 3 5 c d. 2 5 e f g. 2 2 h i j. 2 5 PRE-CALCULUS 11 21

22 Part 5: About You (5 marks) Answer the following questions about your last math course. What was the name of your last math course? Explain the format of the course (e.g., in a classroom, online, print-based distance learning, etc.) When did you finish it? How long did it take you to complete the course? Did you do well in it? Explain: What did you like best about your last math course? Are there certain topics that you find challenging or difficult? 22 PRE-CALCULUS 11

23 What do expect to achieve in PREC 11? Please add anything else about yourself or your previous experiences in math that may help your teacher guide you through this course. /10 Part 1: Linear Equations and Graphing /10 Part 2: Factoring Trinomials /15 Part 3: Quadratics /10 Part 4: Radicals /5 Part 5: About You 50 Total PRE-CALCULUS 11 23

24 24 PRE-CALCULUS 11

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